design of fluid film journal bearings containing...

Design of Fluid Film Journal Bearings Containing Continuous 3D Fluid Pathways which are Formed by Wrapping a Sheet Containing 2D Through-Cut Features

by

Amos Greene Winter, V

B.S., Mechanical Engineering (2003)

Tufts University

Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Mechanical Engineering

at the

Massachusetts Institute of Technology

June 2005

© 2005 Massachusetts Institute of Technology All rights reserved

Signature of Author………………………………………………………………………………… Department of Mechanical Engineering

May 8, 2005

Certified by………………………………………………………………………………………… Martin L. Culpepper

Rockwell International Assistant Professor of Mechanical Engineering Thesis Supervisor

Accepted by………………………………………………………………………………………... Lallit Anand

Chairman, Department Committee on Graduate Students

Design of Fluid Film Journal Bearings Containing Continuous 3D Fluid Pathways which are Formed by Wrapping a Sheet Containing 2D Through-Cut Features

by

Amos Greene Winter, V

Submitted to the Department of Mechanical Engineering on

May 8, 2005 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering

ABSTRACT The purpose of this research was to generate the knowledge required to: (1) design and manufacture fluid film bearings that do not require precision machining processes during fabrication, but rather gain their precision from off-the-shelf parts used in the fabrication process and (2) manufacture parts with 3D internal networks by wrapping thin sheets of material containing 2D through-cut features. This wrapping-based fabrication process, called Three-Dimensional Wrapped Network (3DWN) technology, uses the precision of low-cost, ubiquitous items instead of manufacturing processes to meet the precision requirements of hydrostatic bearings. 3DWN bearings are fabricated by cutting 2D through-cut features into shim stock and then wrapping the shim stock around a precision mandrel. The 2D shim stock features are designed such that they align and form 3D internal networks within the bearing during wrapping. In the final wrapped structure the bore retains the precision diameter of the mandrel and the surface finish of the shim stock, thus meeting the functional requirements of the bearing. This thesis investigates the design and manufacturing of 3DWN hydrostatic bearings. An analytical model was derived to describe the transformation of 3D cylindrical features to 2D through-cut features. Conventional hydrostatic designs and theory were adapted for use in 3DWN bearings. A proof-of-concept was designed, constructed, and tested. Although contact between the shaft and bore was observed during testing, the fluid film stiffness matched theory within 1.6% after accounting for the contact stiffness. The mean bore diameter was measured to be within 0.03% of the mandrel diameter with errors that lie within 5σ of the tolerable error range in the front of the bearing and 2σ in the rear. In a comparison with a conventional hydrostatic bearing of the same size and surface design, the 3DWN cost 10X less. Thesis Supervisor: Martin L. Culpepper Title: Rockwell International Assistant Professor of Mechanical Engineering

BIOGRAPHICAL NOTE

Amos Greene Winter, V was born November 29, 1979 in Peterborough, NH. From Feb –

June 2002, he attended the University of Canterbury in Christchurch, NZ. As part of this

semester abroad, he road his motorcycle through both islands in NZ and solo through the

Australian Outback. He graduated magna cum laude from Tufts University in May 2003

with B.S in Mechanical Engineering. Starting in the fall of that year, he enrolled in the

Massachusetts Institute of Technology and joined the Precision Compliant Systems Lab

(PCSL). This thesis is the culmination of his research in the PCSL. During his masters’

degree, Amos Winter published two conference articles: “Fluid film bearings requiring

no precision machining processes, formed by wrapping 2D sheets.” ASPE 19th Annual

Meeting 2004 and “Design of a gimbaled compliant mechanism stage for precision

motion and dynamic control in Z, θX & θY directions.” ASME DETC 2004. The work

presented in this thesis is currently being prepared for publication in Precision

Engineering.

ACKNOWLEDGMENTS First and foremost I would like to thank Prof. Martin Culpepper for hiring, funding, and

allowing me pursue a project of my own conception. Thank you for striking the

educational balance between advising, motivating, mentoring, and giving me the freedom

to make many discoveries and mistakes on my own. You have made a profound impact

on my life, and I look forward to many fun years ahead during my PhD.

Next, I’d like to thank my labmates Spencer Szczesny and Nate Landsiedel for becoming

two of the greatest friends I have ever made, and supporting me through good times and

bad during my masters. I would also like to thank my other labmates Dariusz Golda,

Shih-Chi Chen, Kartik Mangudi, Soohyung Kim, Rich Timm, Kevin Lin, and Patrick

Carl for your help and support.

The people with whom I am closest in my personal life deserve many thanks. Thank you

Anne, for being so much more than my girlfriend by also being my best friend. Thank

you mom, Lilly, Aunie, and Darlene for your support and encouragement, and providing

a place to getaway and relax. Thank you Alex and Signe, my two lifelong friends who

have had a continuous impact on my life since elementary school. Also I want to thank

Abby, Katie Y, Brian, Chuck, Katie N, John, and Hong for your friendship.

I’d like to recognize the many professors, students, engineers and technicians who added

immense amounts to my education. Thank you Prof. Alex Slocum, Prof. Samir Nayfeh,

Prof. Tim Gutowsky, Mark Belanger, Jerry Wentworth, Maggie Sullivan, Jason Pring,

John Kane, Gil Pratt, for your technical, educational, and inspirational guidance.

Finally on a less serious note, I would like to thank all the people and things that helped

keep me sane through these last two years: My tortoise Nomar, Dave Chappelle, John

Stewart, Jerry Seinfeld, the Boston Red Sox, and my motorcycle.

TABLE OF CONTENTS

ABSTRACT………………………………………………………………………….........3

BIOGRAPHICAL NOTE…………………………………………………………………5

ACKNOWLEDGMENTS....……………………………………………………….……..7

TABLE OF CONTENTS.................................................................................................... 9

LIST OF FIGURES .......................................................................................................... 14

LIST OF TABLES............................................................................................................ 17

1 INTRODUCTION .....................................................................................................19

1.1 Motivation............................................................................................................ 24

1.2 Research Purpose, scope and summary of results ............................................... 27

1.2.1 Questions to be answered in research ......................................................... 28

1.2.2 Research tasks performed ........................................................................... 28

1.2.3 Scholarly contribution of research .............................................................. 29

1.2.4 Summary of results ...................................................................................... 30

1.3 Thesis Organization ............................................................................................. 31

2 BACKGROUND .......................................................................................................32

2.1 Hydrostatic bearings ............................................................................................ 32

2.1.1 How hydrostatic bearings support a load..................................................... 32

2.1.2 Modeling bearing flow................................................................................ 34

2.2 Verification of flat plate assumption in journal bearings..................................... 36

2.3 Means of fluid restriction..................................................................................... 38

2.4 Surface self-compensated bearings...................................................................... 39

3 3DWN BEARING DESIGN......................................................................................43

3.1 Inception of 3DWN technology............................................................................ 43

3.2 Early 3DWN bearing prototypes .......................................................................... 45

3.3 Motivation to design a HBP................................................................................. 45

3.4 Satisfying precision requirements of the HBP..................................................... 46

3.4.1 Characterization of surfaces........................................................................ 47

3.4.2 Satisfying surface finish requirements........................................................ 48

3.4.3 Satisfying bore diameter and roundness requirements ............................... 49

3.5 Design of 3DWN HBP bore surface features ...................................................... 49

3.5.1 Inspiration for HBP surface feature design................................................. 50

3.5.2 HBP pad configuration ............................................................................... 51

3.5.3 Overlap region of bearing bore ................................................................... 53

3.5.4 Restrictor design ......................................................................................... 54

3.5.5 Full 3DWN HBP bore feature design ......................................................... 56

3.6 Design of Internal Channels................................................................................. 57

3.6.1 2D through-cut parameters ......................................................................... 58

3.6.2 Design of HBP fluid networks for low resistance ...................................... 59

3.6.3 Feed channel design.................................................................................... 59

3.6.4 Cross-connection channel design................................................................ 59

3.6.5 Drainage channel design ............................................................................. 60

3.7 Summary.............................................................................................................. 60

4 MODELING AND ANALYSIS................................................................................61

4.1 Wrapping model................................................................................................... 61

4.1.1 Describing a wrapped structure .................................................................. 61

4.1.2 3D to 2D coordinate transformation ........................................................... 65

4.2 Modeling bearing performance............................................................................ 70

4.2.1 Fluid resistance modeling ............................................................................ 70

4.2.2 Resistance ratio ............................................................................................ 72

4.2.3 Derivation of effective pad area................................................................... 73

4.2.4 Derivation of bearing stiffness..................................................................... 75

4.3 Sensitivity Analysis .............................................................................................. 76

4.3.1 Justification for using non-precision cutting processes .............................. 76

4.3.2 Sensitivity of performance to internal channel errors................................. 78

4.3.3 Justification for neglecting tension in the wrapping model ........................ 80

4.3.4 Sensitivity to bore bulge to channel placement .......................................... 81

4.3.5 Appropriate wrapping tension to compress Template deformities ............. 82

4.4 Summary............................................................................................................... 83

5 MANUFACTURING A 3DWN HBP .......................................................................84

5.1 Failed attempts at adhering wrapped layers......................................................... 84

5.2 Template fabrication ............................................................................................ 86

5.2.1 Waterjet cutting........................................................................................... 86

5.2.2 Fixturing of Template within waterjet ........................................................ 87

5.3 Template wrapping .............................................................................................. 87

5.3.1 Rolling jig ................................................................................................... 87

5.3.2 Alignment of Template to mandrel............................................................. 88

5.3.3 Adhesion of wrapped layers........................................................................ 90

5.4 Packaging the Template in a housing .................................................................. 91

5.4.1 Joining Template and housing .................................................................... 91

5.4.2 Prepping the HBP for casting ..................................................................... 92

5.4.3 Casting the Template into the housing........................................................ 93

5.4.4 Finishing procedures................................................................................... 94

5.5 Summary............................................................................................................... 95

6 EXPERIMENTAL VERIFICATION........................................................................96

6.1 Experimental setup............................................................................................... 96

6.5.1 Parameters of the 3DWN HBP used in experimentation............................ 96

5.1.2 Experimental setup for stiffness testing...................................................... 98

5.1.3 Bore measurement ....................................................................................... 99

5.2 Stiffness results and discussion........................................................................... 100

5.2.1 Stiffness test results.................................................................................... 100

5.2.2 Sources of error in stiffness data................................................................ 101

5.3 Results from bore measurements ........................................................................ 104

5.4 Cost comparison.................................................................................................. 104

5.5 Summary............................................................................................................. 105

7 SUMMARY.............................................................................................................106

7.1 Scholarly Contributions ..................................................................................... 106

7.2 Engineering impact ............................................................................................ 107

7.3 Future work........................................................................................................ 108

REFERENCES................................................................................................................110

LIST OF FIGURES

Figure 1.1 3DWN bearing manufacturing process ........................................................... 21

Figure 1.2 Precision requirements decoupled from fabrication of the bearing................ 22

Figure 1.3 Finished 3DWN bearing.................................................................................. 23

Figure 2.1 Configurations and pressure profiles for different bearings........................... 33

Figure 2.2 Fluid relationships in hydrostatic bearings..................................................... 34

Figure 2.3 Velocity profile of fully developed flow........................................................ 34

Figure 2.4 Bearing eccentricity during shaft displacement ............................................. 37

Figure 2.6 HydroglideTM surface self-compensated bearing [1]...................................... 41

Figure 2.7 Fluid circuit for surface self-compensated bearing ........................................ 41

Figure 3.1 Flat actuator concept and implementation...................................................... 44

Figure 3.2 First 3DWN mock-up and rolling process...................................................... 44

Figure 3.3 Example surface roughness profile [20]......................................................... 47

Figure 3.4 Determination of Ra value [21]...................................................................... 48

Figure 3.5 Hydrostatic self-compensated journal bearing [22] ....................................... 50

Figure 3.6 Annular restrictor designs for hydrostaic surface self-compensated bearings 51

Figure 3.7 Comparison of pad configurations ................................................................. 52

Figure 3.8 Chosen 3DWN HBP bore surface features layout ......................................... 53

Figure 3.9 Geometric matching of overlap region........................................................... 54

Figure 3.10 Single feed, double annulus restrictor configuration.................................... 55

Figure 3.11 3DWN self-compensated bearing pad bore surface features and geometric

parameters ................................................................................................................. 56

Figure 3.12 3DWN self-compensated bearing template................................................... 58

Figure 4.1 Diagram of overlap region of layer one and two............................................ 62

Figure 4.2 Model used for x-displacement of cantilevered beam.................................... 64

Figure 4.3 Transformation of 3D cylindrical features to Template features ................... 66

Figure 4.4 Local coordinate system for position along cantilevered beam ..................... 67

Figure 4.5 Flow over bearing bore surface features ........................................................ 71

Figure 4.6 Fluid circuit of one set of opposed pads in the HBP ...................................... 71

Figure 4.7 Pressure profile over bearing pad ................................................................... 73

Figure 4.8 Pad stiffness configuration of HBP (gap greatly exaggerated) ...................... 75

Figure 4.9 Sensitivity of the HBP stiffness to manufacturing errors for h/R = .002, h/l =

0.011.......................................................................................................................... 77

Figure 4.10 Channel constriction as a result of internal feature misalignment ............... 78

Figure 4.11 Internal channel resistance sensitivity to expected error range of t and R for

t/R = 0.01................................................................................................................... 79

Figure 4.12 Error caused by wrapping tension ................................................................. 81

Figure 4.13 Bore bulge resulting from pressurized channels .......................................... 81

Figure 4.14 Deflection of Template due to tension ......................................................... 82

Figure 4.15 FEA Results from deformity deflection under tension ................................ 83

Figure 5.1 Template being cut in waterjet ....................................................................... 86

Figure 5.2 Kinematic fixture for waterjet cutting, waterjet cutting setup........................ 87

Figure 5.3 3DWN bearing rolling jig............................................................................... 88

Figure 5.4 Template mounted on rolling jig .................................................................... 89

Figure 5.5 Adhering adjacent layers within the HBP ...................................................... 91

Figure 5.6 Centering of Template within housing ........................................................... 92

Figure 5.7 Preparations for casting .................................................................................. 93

Figure 5.8 Wrapped Template cast in housing ................................................................ 94

Figure 5.9 Drainage ports with grease plugs removed .................................................... 94

Figure 5.10 Residual super glue to be removed from bearing bore surface features....... 95

Figure 6.1 Experimental setup for testing stiffness .......................................................... 98

Figure 6.2 Oil pressurization device ................................................................................ 99

Figure 6.3 Bore precision testing using a CMM............................................................ 100

Figure 6.4 Measured stiffness vs. theory ....................................................................... 101

Figure 6.5 View of leakage flow (shaft removed from bore) ........................................ 102

Figure 6.6 Bearing fluid circuit including leakage flow................................................ 102

Figure 6.7 Theoretical stiffness with and without leaks ................................................. 103

LIST OF TABLES

Table 1.1 Comparison of different bearing types [2]....................................................... 24

Table 1.2 Hydrostatic journal bearing applications [1-3] ................................................ 25

Table 2.1 Methods of bearing compensation................................................................... 39

Table 3.1 Progression of early 3DWN prototypes........................................................... 45

Table 5.1 Failed attempts at adhering wrapped layers..................................................... 85

Table 6.1 Parameters of the 3DWN HBP used in experimention.................................... 97

Table 6.2 Measurements of bore at front and rear of bearing........................................ 104

Table 6.3 Cost comparison 3DWN and conventional bearing ...................................... 105

CHAPTER

1 INTRODUCTION

The purpose of this research was to generate the knowledge required to:

1. Design and manufacture fluid film bearings (FFBs) that do not require precision

machining processes during fabrication, but rather gain their precision from off-

the-shelf parts used in the fabrication process.

2. Manufacture parts with 3D internal networks by wrapping thin sheets of material

containing 2D through-cut features.

All FFBs support a load on a pressurized film of fluid. The bearing surfaces have to be

precise in their spacing and surface finish to insure uniform film properties and to insure

no mechanical contact between bearing components. Traditional FFBs require precision

machining processes to provide the requisite geometric accuracy and surface finish.

Additionally, fluid static bearings require internal networks to distribute fluid throughout

the bearing. Current FFBs have the following drawbacks which limit their use in

widespread engineering applications:

• Precision machining processes used to make the bearing surface contribute

significantly to the overall cost of the bearing.

• Fluid networks have to be machined or cast into the bearing, which adds extra

labor time, manufacturing steps, and cost to the bearing.

• Fixed restrictor bearings require tuning for optimum performance (means of

restriction are discussed in a later section), adding further labor time and effort

during bearing installation.

• Self-compensated bearings, which self-tune and can achieve twice the stiffness of

fixed restrictor bearings [1], require extensive internal channels which connect

restrictors and pads.

The central thesis of this research is that FFBs can be fabricated with incorporated

internal networks by wrapping thin 2D sheets. Further, this may be accomplished

without the need for a bearing manufacturer to perform precision machining processes.

This wrapping-based fabrication process, called Three-Dimensional Wrapped Network

(3DWN) technology, uses the precision of low-cost, ubiquitous items instead of

manufacturing processes to meet the precision requirements of the bearing. Figure 1.1

illustrates how 3DWN bearings are made.

Step Process

1

Precision diameter

ground mandrel

Precision surface finish

and thickness material

(e.g. shim stock)

Precision diameter

ground mandrel

Precision surface finish

and thickness material

(e.g. shim stock)

2 Template cut into shim stock with 2D process

(e.g. laser cutting)

Adhesive applied to back side

Template cut into shim stock with 2D process

(e.g. laser cutting)

Adhesive applied to back side

3 Template wrapped about mandrel

Adhesive bonds

each layer

Template wrapped about mandrel

Adhesive bonds

each layer

Fin

ish

3DWN

bearing

3DWN

bearing

Precision

surface finish

from shim

stock

Precision

diameter from

mandrel

Precision

surface finish

from shim

stock

Precision

diameter from

mandrel

3D internal

features

formed by

template

3D internal

features

formed by

template

Figure 1.1 3DWN bearing manufacturing process

The 3DWN process begins with two off-the-shelf parts that have inherent precision:

1. A precision ground mandrel with the required diameter size, tolerance, and

circularity for the bearing

2. Cold rolled shim stock with the required surface finish for the bearing

A pattern of through-cut features is cut into the shim stock, forming a template. Adhesive

is applied to the back of the template. The features in the template are designed such that

they form internal networks within the bearing when the template is wrapped. In its final

form, the bearing bore retains the precision diameter of the mandrel and the surface finish

of the shim stock, thus meeting the functional requirements of the bearing.

3DWN bearings can be made less expensively than conventional hydrostatic journal

bearings given that the precision requirements of the bearing are satisfied by low-cost,

well-developed manufacturing processes. The chart in Figure 1.2 demonstrates how the

precision of the bearing is decoupled from the fabrication of the bearing.

Off-the-shelf shim

stock with cold rolled

precision surface finish

Off-the-shelf precision

ground mandrel

Precision parts

Off-the-shelf shim

stock with cold rolled

precision surface finish

Off-the-shelf precision

ground mandrel

Precision parts

Wrapping

Waterjet or laser

cutting

Non-precision

processes

Wrapping

Waterjet or laser

cutting

Non-precision

processes

[1,6]

When wrapped, 2D cut

features in adjacent layers

align to form internal

networks

Fluid networked between

input, restrictors, pads, and

drains

Bearing features cut into

Template using 2D through-

cutting process

Restrictors, pads, and drains

cut in bore surface

Template, made from shim

stock with precision surface

finish, forms bore surface

The maximum peak-to-

valley surface roughness not

greater than one-fourth the

bearing gap

Template wrapped around

precision ground mandrel,

replicating mandrel diameter

Design ParametersFunctional Requirements

When wrapped, 2D cut

features in adjacent layers

align to form internal

networks

Fluid networked between

input, restrictors, pads, and

drains

Bearing features cut into

Template using 2D through-

cutting process

Restrictors, pads, and drains

cut in bore surface

Template, made from shim

stock with precision surface

finish, forms bore surface

The maximum peak-to-

valley surface roughness not

greater than one-fourth the

bearing gap

Template wrapped around

precision ground mandrel,

replicating mandrel diameter

Total diameter errors

within one-forth of

bearing gap

Design ParametersFunctional Requirements

Figure 1.2 Precision requirements decoupled from fabrication of the bearing

3DWN technology enables greater flexibility in design because many sizes of 3DWN

bearings can be made with the same process. For instance, a 3DWN bearing assembly

line equipped with one wrapping machine and a variety of mandrel sizes could produce

multiple bearing designs and sizes. Mandrels could be re-used between bearings, which

could further reduce the cost of production.

A self-compensated hydrostatic journal bearing, shown in Figure 1.3, was designed,

modeled, and tested as a case study for 3DWN technology. This bearing is composed of

two sets of four radial pads, which give the bearing moment and radial stiffness. The

template contains all bearing features and internal fluid networking, including feed, drain,

and cross channeling between the restrictors and pads. The 3DWN bearing is potted

within an aluminum sleeve. In this form, the quasi-monolithic assembly is structurally

stiff. The housing allows for mounting and provides a connection to pressurized fluid.

Restrictor

Pad Pocket

Drainpocket

Wrappedbearing

Pottingcompound

Aluminumsleeve

A. Bearing after wrapping B. Bearing cast into an aluminum housing

Figure 1.3 Finished 3DWN bearing

The bearing shown in Figure 1.3 is a proof-of-concept, and is not designed for any

specific application. This thesis presents the design process, modeling, and fabrication

used to make the prototype shown in Figure 1.3. The methods described may be used by

engineers to design 3DWN bearings for specific applications.

1.1 Motivation

Hydrostatic bearings are utilized in applications that require high load capacity (on the

order of mega Newtons), high stiffness (N/nm) [1], and low friction without stick-slip.

These bearings have advantages over other types of journal bearings. A comparison of

different bearing types is presented in Table 1.1. The bearings are rated 1 to 5, with high

being most favorable.

Table 1.1 Comparison of different bearing types [2]

Types

Characteristics

Design 3 3 4

Positioning accuracy 2 2 2

Assembly 3 3 2

Cost (to manufacture) 3 2 3

Cost (to install) 3 2 4

Life 3 4 3

Lubrication circuit 3 2 4

Cost of lubrication circuit 3 2 4

Supply pressure and pumping power 3 2-3d 4

Load 2-3b

2-4d 3

Stiffness 2-3b

2-4d 3

Vibration damping 2-3c 4 2

Friction coefficient and frictional power 3 3-5b 4

Stick-slip 4 5 5

Wear 3 5 3

Notes: a

Fixed Restrictor, b

Depends on Speed, c

Whirl, d

Depends on Supply Type

Considerations for journal bearings

Hydrodynamic Hydrostatica

Rolling Elements

A hydrostatic journal bearing uses a high pressure fluid film to support a shaft.

Incompressible fluid is used and a large portion of the bearing bore is pressurized, which

results in high forces and stiffness. There are no mechanical interactions between the

shaft and the bearing bore since the shaft is completely supported on the fluid film. As a

result there is no stick-slip, making the motion of the bearing highly repeatable.

Hydrostatic bearings are ideal for applications where high load capacity, high stiffness,

and low friction are needed. Unlike hydrodynamic bearings, hydrostatic bearings do not

require a spinning shaft to maintain their load-bearing properties. As such, they have

excellent performance in stationary and stop-start operation. Table 1.2 lists specific

applications and benefits of hydrostatic journal bearings.

Table 1.2 Hydrostatic journal bearing applications [1-3]

Telescopes

Radio telescopes

Radar antennas

Air preheaters for boilers

Rotating mills for ores or slags

Large boring machines

Large milling machines

Large lathes

Assembly lines

Large structures

Grinding machines

CNC machining centers

Medium-high velocity spindles

Precision balances

Dynamometers

Vibration attenuators

Frictionless oil seals

Small machines

Medium sized machines

Large machines

High load capacity

No wear

Precision rotation

Squeeze film damping

High stiffness

No stick-slips

Lubricated while stopped

No wear

Near zero friction at low

speed

Categories Specific applications Benefits

Hydrostatic journal bearing applications

Hydrostatic journal bearings offer better performance than other types of bearings.

However, there are factors that make them undesirable for some applications:

• Bore machining cost: Traditional hydrostatic journal bearings require a precision

process to machine the bore to the required diameter and surface finish.

• Fluid channel machining cost: Hydrostatic bearings require fluid to be injected

into multiple locations between the bore and shaft. Distribution of fluid requires

an internal network system around the bearing.

• Added machining for self compensation: Self compensating bearings offer higher

load capacities and stiffness than fixed compensator bearings, but require

additional fabrication to produce cross-linked fluid pats for restrictors and pads.

• Custom design and low production volume: Hydrostatic bearings can vary in and

size and surface feature layout, requiring most designs to be customized and made

in small numbers.

Methods have been employed to lower the cost of producing hydrostatic journal bearings,

yet all of these require at least one precision machining process. Self-compensated

hydrostatic journal bearings, made by Kotilainen, et al. [4,5], were formed by sand or

investment casting. Although these bearings require fewer manufacturing steps than

conventional FFBs, the bore requires post-casting precision machining post and the

molds are destroyed during production, which further adds cost.

Kotilainen’s cast bearings are a spin-off from a reduced-cost technology, the

TurboToolTM and HydroSpindleTM [6-9], which have the bearing features cut into the

shaft instead of the bearing bore. These technologies still require 3D machining and

precision grinding of the shaft. Another method of making hydrostatic bearings is by

pressing bronze sleeves into a block. In this process the bore has to be post machined to

compensate for the press fitting [10]. Also, Babbitting, where molten metal is cast around

a conical shaft, has been used for decades to make replicated bores, but this process also

requires finish machining [10, 11]. Polymers have been through-cut with a bearing bore

surface features and adhered to bearing bores [Lyon patent], but this process requires the

bore to be machined to same diameter precision as a conventional hydrostatic bearing.

Kotilainen’s research demonstrates that an imprecise process such as casting may be used

to make the bearing bore surface bore surface features without adversely affecting

performance. This reduces the cost of producing these bearings below that of

conventional bearings [4,5]. In a similar fashion 3DWN technology uses imprecise

through-cutting processes like laser or waterjet cutting to make the bearing features. As

such, the precision requirements may be met by off-the-shelf parts. The advantages of

3DWN technology in bearing manufacturing are summarized below.

• Decoupled precision: The precision requirements of the bearing are

compartmentalized within the mandrel and shim stock, which are low cost and

easy of others to fabricate. Cutting of the template does not require a precision

process.

• Included feed channels: Bearing features and internal network features are all

included in the template.

• 1-Step machining: Using 3DWN technology, the bearing features and the network

features are cut in the template at the same time using the same process.

• Monolithic construction: 3DWN wrapped structures are made from one part, and

may easily be potted into a housing to make a bearing.

• Flexible production: The mandrel is not destroyed during the wrapping process.

Thus, a single wrapping system with multiple mandrels may make varying

bearing sizes.

1.2 Research Purpose, scope and summary of results

1.2.1 Questions to be answered in research

The questions that are answered through this research are:

1. How can a cylindrical structure with internal features be modeled as a wrapped

structure, and how is the wrapped structure modeled unrolled as a 2D sheet with

through-cut features? What errors result that affect bearing performance result

from this process?

2. How is the level of precision quantified for the off-the-shelf parts used in 3DWN

bearing fabrication?

3. How are 3DWN bearings manufactured? What bearing materials, adhesives, and

support structures have to be considered in implemented. What is the best method

for rolling?

4. What are the practical issues regarding the design process, fabrication, and

implementation of 3DWN technology to fluid film bearings?

5. How can a 3DWN bearing design be chosen for a particular application? What

modeling techniques are necessary to select a 3DWN bearing design, determine

its dimensions, and predict its performance?

1.2.2 Research tasks performed

These questions were answered through the following research tasks:

1. A model to transform 3D cylindrical coordinates to 2D sheet coordinates was

derived. Errors were introduced into the model and resulting sensitivity to bearing

performance was analyzed.

2. Conventional bearing theory for precision requirements was used to form metrics

for off-the-shelf precision parts.

3. Materials suitable to wrapping and use in the bearing were determined. Multiple

adhesives were tested. Prototypes to verify manufacturing methods were built.

4. Current fluid film bearing theory and bearing feature designs were modified for

use in 3DWN bearings.

5. A 3DWN bearing prototype was built and tested as a bench-level prototype.

1.2.3 Scholarly contribution of research

The following scholarly contributions are a result of the work presented in this thesis:

1. 3DWN technology is a new method of making a precision bearing bore. Surface

replication by wrapping is a deviation from conventional bearing manufacturing

practices; all current fluid film journal bearings require at least one precision

process in their construction.

2. In the 3DWN manufacturing process the errors associated with the non-precision

parts have less of an effect on bearing performance then errors in the precision

parts. This is a powerful relationship, in that it supports the feasibility of 3DWN

technology by showing a hydrostatic bearing’s precision requirements can be

decoupled from the fabrication processes used to make it.

3. Metrics to judge the level of precision required in the off-the-shelf parts used to

make a 3DWN bearing are defined from conventional hydrostatic bearing theory.

4. A new method of manufacturing 3D parts with internal features by using 3DWN

technology. 3DWN is not limited to fluid channels; it could be used for many

kinds of applications that require cylindrical internal networks.

5. A model for describing a wrapped structure is derived. This model enables

coordinates in a 3D cylindrical structure to be transformed to a 2D Template.

When the Template is rolled, the 2D features align in the wrapped structure to

recreate the original 3D features.

6. Existing bearing features are modified such that it can be cut into a 2D Template.

Conventional hydrostatic bearing analysis is used to evaluate the bearing

performance.

1.2.4 Summary of results

The final 3DWN bearing prototype designed, constructed, and tested for this thesis was a

surface self-compensated hydrostatic journal bearing. The bearing had a bore surface

features consists of two axial sets of four circumferentially spaced pads, each connected

to an opposed surface restrictor. It was lubricated with heavy weight motor oil and

pressurized with shop air regulated to a safe level (100psi) for laboratory experiment.

Load capacity and stiffness were tested by applying varying loads to the shaft and

measuring the eccentricity of the shaft within the bearing.

Although contact between the shaft and bore was observed, the fluid film stiffness

matched theory within 1.6% after accounting for the contact stiffness. The mean bore

diameter was measured to be within 0.03% of the mandrel diameter with errors that lie

within 5σ of the tolerable error range in the front of the bearing and 2σ in the rear. In a

comparison with a conventional hydrostatic bearing of the same size and surface design,

the 3DWN cost 10X less.

1.3 Thesis Organization

Chapter 2 presents a background on hydrostatic bearings, which includes modeling of

bearing fluid flow and means of restriction. The third chapter describes how the concept

for 3DWN bearings was conceived and how 3DWN bearings are designed. The fourth

chapter presents the wrapped structure model, error functions, the use of established

bearing theory in 3DWN bearings, and a cost comparison with conventional hydrostatic

journal bearings. The fifth chapter describes the materials and adhesives were chosen and

how 3DWN bearing prototypes were constructed. The experimental setup, testing

procedure, and results are presented in the sixth chapter. Possible sources of error are also

identified and verified through testing and examination. The seventh chapter provides a

summary of results and a discussion of future research.

CHAPTER

2 BACKGROUND

This chapter covers the basic theory behind hydrostatic bearings. The first section

provides an explanation of how a hydrostatic bearing supports a load. The second section

presents verification the flat plate model used in journal bearings. The third section

describes methods of restricting bearing flow. The final section gives a background on

surface self-compensated bearings.

2.1 Hydrostatic bearings

2.1.1 How hydrostatic bearings support a load

Hydrostatic bearings support a load on a thin film of pressurized fluid. The origins of

these bearings can be traced to L.G. Girard, who in 1852 made the first water hydrostatic

journal bearing [13]. In all hydrostatic bearings, pressurized fluid is pumped into the pad

and slowly flows out over the bearing lands. The bearing gap is small enough to restrict

the flow over the lands and induce a linear pressure drop from viscous losses. Depending

on whether or not the bearing is pocketed, the pressure profile over the bearing surface

has a triangular or trapezoidal pressure profile, as shown in Figure 2.1. A pocketed

bearing has the benefit of a larger area exposed to input pressure, thus resulting in higher

load capacity and stiffness

Qin

Pin

Qin

Pin P

in

Qin

Pin

Qin

Pin

Pa

Pin

Pin

Pa

Pin

Pa

Pin

Pa

A. Pocketed bearing B. Non-pocketed bearing

Figure 2.1 Configurations and pressure profiles for different bearings

In order for the bearing to support a load and have stiffness, the flow must be restricted

before it enters the bearing pad. This may be visualized with the electrical analogy shown

in Figure 2.2A. In this analogy Ohm’s Law of V = IR is replaced with P = QR. Here P is

the pressure, Q is the volumetric flow rate, and R is the resistance to fluid flow. For this

example RR is the restrictor resistance and has a fixed value. The fluid resistance caused

by the fluid flow over the bearing lands, RP, increases as a function of decreasing gap

height cubed. Because PS is fixed, as the bearing gap is decreased RP increases, thus

increasing the pressure drop over the lands and raising the pressure in the bearing pocket.

This phenomenon can be seen in Figure 2.2B. As a load, W, is applied, the gap height

decreases, creating a taller trapezoidal pressure profile, which results in a larger reaction

force, giving the bearing stiffness.

RR

RP

PATM

PS

RR

RP

PATM

PS

h

Q > 0

F

Q > 0

F + dF

h - δ

Pa

Pin

Pa

Pin

A. Electrical analogy for bearing fluid network

B. Pressure profiles from changes in gap height [4, 2]

Figure 2.2 Fluid relationships in hydrostatic bearings

2.1.2 Modeling bearing flow

Using lumped parameter modeling, most of the regions of hydrostatic bearings

(with the exception of some types of restrictors) may be modeled as one dimensional,

laminar, fully developed flow between two plates, as shown in Figure 2.3. The flow may

be considered laminar as viscous effects dominate through the small bearing gap to create

flow restriction, and the Reynold’s number is << 2000. The flow may be considered

locally fully developed because h << L [14].

y

x

z

hu

y

x

z

hu

Figure 2.3 Velocity profile of fully developed flow

The resulting Navier-Stokes equation for one-dimensional flow is given by Equation

(2.1).

(2.1)

Equation (2.1) can be reduced using the following assumptions:

1. During operation under constant supply pressure, the flow is steady.

2. Since hydrostatic bearings use incompressible fluids, mass conservation though a

uniform bearing gap requires that the velocity in the x direction be constant.

3. There can not be any flow through the walls of the bearing gap, so there is no

flow in the y direction.

4. The flow is uniform and does not vary in the z direction.

5. Horizontal height changes are negligible, so body forces can be ignored.

The reduced Navier-Stokes equation is given by Equation (2.2).

dx

dp

dy

ud=

2

2

µ (2.2)

Integrating Equation (2.2) twice and applying the non-slip boundary conditions of

u(0)=u(h)=0, yields Equation (2.3) for the flow velocity.

xgx

pu

zu

yu

xz

uw

y

uv

x

uu

t

uρµρ +

∂

∂−

∂

∂+

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2

1 2 3 4 2 4 5

)(2

1yhy

dx

dpu −

=

µ (2.3)

Integrating the velocity over the entrance area of the bearing, which is defined by the

height and width of the land, results in Equation (2.4) for the volumetric flow rate.

=

dx

dphwQ

µ12

3

(2.4)

Where w is the width of the land. To express pressure, flow rate, and flow resistance in

the analogous form of V = IR, the pressure can be integrated over the land length L to

obtain Equation (2.5).

wh

L

Q

pR

3

12µ=

∆= (2.5)

This expression is powerful because it can be used to express all the geometric

parameters of the bearing as a fluid resistance. As such, the bearing may be modeled with

lumped parameters, with each parameter being composed of variations of Equation (2.5).

2.2 Verification of flat plate assumption in journal bearings

A problem arises when using the flat plate approximation for journal bearings: journal

bearings are not flat. Figure 2.4 shows a displaced shaft within a journal bearing. As a

result of the curvature of the bearing, the gap does not decrease uniformly. Modeling

flow in a uniform gap is much simpler than in a varying gap, thus a flat plate

approximation is advantageous if it doesn’t deviate significantly from the curved plate

model. This section will investigate the error associated with using the flat plate

approximation instead of a curved plate model. The method used was adapted from [4,6]

δ

θx

pinflo

w

Bearing Shaft

δ

θx

pinflo

w

Bearing Shaft

Figure 2.4 Bearing eccentricity during shaft displacement

The gap height, h, is defined by the expression given in Equation (2.6).

+−=

D

xhh o θε cos1 (2.6)

Where ho is the initial gap height before deflection, θ is the initial angular position of the

height being measured, x is the arch length position on the face of the bearing, D is the

diameter of the bearing, and ε is the eccentricity ratio defined by Equation (2.7).

oh

δε = (2.7)

By substituting Equation (2.6) into Equation (2.4), the pressure drop along the arch length

is expressed by Equation (2.8).

3

3cos1

112

+−

−=

D

xh

w

Q

dx

dp

o θε

µ

(2.8)

the ratio L

x=ξ may be used to find the pressure drop along the arch length from

Equation (2.8) using a definite integral. The resulting flow resistance in the

circumferential direction is given by Equation (2.9).

∫−

+−

=2

1

2

133

cos1

112ξ

ξθε

µd

D

Lwh

LR

o

(2.9)

It is important to note that the ratio L/D is different than the usual L/D ratio that

corresponds to the bearing length and diameter. In Equation (2.9) L/D is the ratio of the

land length and the diameter. Dividing Equation (2.9) by Equation (2.5) and evaluating at

L/D = 0.1, which is a realistic approximation [4], and an eccentricity of 0.5, which

corresponds to the operating range of a typical bearing [2], the ratio of resistances at any

radial position around the pad, θ = 0o to 90o, is very small (<1%). Thus, the flat plate

approximation may be used to accurately model fluid film journal bearings.

2.3 Means of fluid restriction

As mentioned previously, restrictors are required in bearing fluid circuits. They induce a

pressure drop in the fluid as it enters the bearing pad. When the resistance of the bearing

lands increases, the pressure in the bearing pad increases. The bearing has stiffness

because the load capacity increases with bearing displacement.

Table 2.1 describes different types of restrictors that are commonly used. This table is a

summary of information on restrictors from [1,2].

Table 2.1 Methods of bearing compensation

Direct feed, constant

flowrate

This type of compensator provides a fixed flowrate.

Pressure in the pads is increased by restricting the flow

over the bearing lands, so as the bearing is displaced the

pressure in the pads increases. The max pressure attainable

is limited by the power of the motor pump.

Capillary tubes

This type of compensator uses viscous losses from laminar

flow. The restrictor is usually made from a capillary tube.

The length of the tube is adjusted to tune the restriction.

Orifices

Orifices cause a pressure drop by forcing the fluid around a

sharp corner and choking the flow. This type of

compensation is more common in aerostatic bearings

because of the low viscosity of the air. They are less

desirable for precision applications because they induce

turbulent flow.

Proportional flow

restrictors

These restrictors are often made by splitting the flow into

opposing bearing lands and forcing it over a compliant

diaphragm. When there is a pressure difference between the

lands, the diaphragm will stretch to lower the flowrate to

the pocket with lower pressure. This lowers the opposing

force against the bearing supporting more load, thus making

the bearing pair stiffer.

Surface self-compensated

bearings

This method of restriction works by forcing the fluid

through passages on the bearing surface. Each restrictor is

connected to an opposing pad. The main advantage of these

compensators is that there are no small channels in which to

trap particles, so clogging is much less of a problem. This

compensation method is fully explained below.

Compensator type Description

2.4 Surface self-compensated bearings

The following is a summary of surface self-compensated bearings from [1,3,15,16]. In

these bearings the restrictors are located on the bearing surface. A pressure drop is

induced by viscous losses in the flow through the bearing gap. Self-compensation

provides the following advantages over other types of hydrostatic bearings.

1. Variable restrictors – Since restriction is accomplished on the bearing surface, the

restrictor resistance changes with bearing displacement. If the restrictors are in an

opposed configuration, shown in Figure 2.5A and Figure 2.6, greater pad

pressures can be achieved. For example when the bearing is displaced down, the

upper restrictor resistance (RUR) will decrease and the lower pad resistance (RLP)

will increase, raising the pressure and vertical force of the lower pad. If properly

designed, these bearings can exhibit stiffnesses that are twice as high as fixed

compensator bearings of the same pad size [1]

2. No tuning required – Self-compensated bearings automatically adjust the force

between the pads and tend towards a stable neutral position. As a result no tuning

is required to balance the pads.

3. Large fluid channels – The smallest features in the fluid circuit are on the bearing

land. This makes self-compensated bearings much less susceptible to clogging

than bearings that use small clearance restrictors, such as orifices and capillary

tubes. The fluid flow over the restrictors and lands constantly cleans the bearing

surface.

A. Fluid circuit of opposed pads B. HydroglideTM topology

Figure 2.5 HydroglideTM surface self-compensated bearing [1]

Ps

RUR

RLPRLR

RUP

RLL

RUL

PA

Figure 2.6 Fluid circuit for surface self-compensated bearing

The HydroglideTM, shown in Figure 2.5, is a unique design in self-compensated bearings

because the resistance of the restrictor is deterministic. Fluid flows from an outer annulus

radially inwards to a center hole, shown in Figure 2.5A. This configuration is different

from early restrictor designs, where fluid flows outwards from a source to collection

pockets [3,16]. The HydroglideTM is easier to model than other self-compensated

bearings because the bearing flow is in the opposite direction of leakage flow, making

leakage not affect bearing performance.

CHAPTER

3 3DWN BEARING DESIGN This chapter presents the process used to design the 3DWN HBP. The chapter begins

with the inception and evolution of 3DWN technology then focuses on the design of the

most recent 3DWN surface self-compensated HBP. The design process is composed of

three stages: 1) satisfying the precision requirements of the HBP, 2) modifying existing

bearing technology for incorporation into a wrapped structure, and 3) designing the

remainder of the Template to include internal networks. Qualitative design choices based

upon sound engineering knowledge compose the majority of this chapter. Quantitative

methods behind design decisions are mentioned in this chapter with the full analysis

explained in Chapter 4.

3.1 Inception of 3DWN technology

3DWN technology grew out of research on flat pneumatic and hydraulic actuators. The

initial objective of that project was to research the feasibility of making flat, monolithic

pneumatic and hydraulic actuators with pistons, actuator bodies, return spring

mechanisms, and fluid routing channels. The actuators were designed to have fluid film

bearings on the piston surfaces to reduce friction, as shown in (Figure 3.1 A). The bearing

gaps were created by compliant end caps, which expanded when the piston was

pressurized. A goal of the project was to enable the production of monolithic robot

structures, shown in Figure 3.1 B, which could be low-cost, expendable, and used for

educational or military applications.

Compliant Return Spring

Air

Bearing

Flexure

Seal

Actuator

Body

Washer

Plate

Compliant End Cap

Piston

Compliant Return Spring

Air

Bearing

Flexure

Seal

Actuator

Body

Washer

Plate

Compliant End Cap

Piston

A. Flat actuator prototype with integrated air bearing B. FlatBot concept

Figure 3.1 Flat actuator concept and implementation

Engineering flat actuators presented many challenges. Sealing the piston was near

impossible as it had sharp corners and the compliant end caps bowed under pressure. The

end caps bowing instead of deflecting uniformly creates a parabolic bearing gap. The

restriction of the bearing gap scales with the gap height cubed, so small errors in gap

height are detrimental.

To overcome sealing issues the author conceived of a conventional cylindrical piston

which slides on aerostatic journal bearings. The internal channels and features of the

bearings would be made of cylindrical layers, similar to the flat layers of the flat

actuators.

After some consideration the author realized that the cylindrical layers could be made

from one sheet wrapped around the piston. Each layer could contain 2D cutout features

which would align to form 3D channels when the sheet was wrapped. The first model of

3DWN technology, shown in Figure 3.2, was made from paper.

Figure 3.2 First 3DWN mock-up and rolling process

3.2 Early 3DWN bearing prototypes

The 3DWN aerostatic bearings shown in Table 3.1 were constructed as the first proof–of-

concept prototypes. Each of these prototypes was made from plastic shim stock that was

adhered with double-stick tape. The first bearing prototypes were aerostatic instead of

hydrostatic. A compressed air instead system was more convenient for bench-level

testing, as air can be directly regulated from storage tanks. The information that was

learned through these prototypes as well as the difficulties associated with each

prototype, is presented in Table 3.1.

Table 3.1 Progression of early 3DWN prototypes

a. Flat actuators not working – deformation = leaks

b. Round actuators with incorporated wrapped bearing

c. Wrapped structure for internal channels

Origin of

idea1

a. Wrapped Template features alignb. Internal networks carry fluid

c. Compliant Bodyd. Feeding channels warp bearing surface

First air bearing

prototype2

a. Casting into housing improves body stiffnessb. Plastic too compliant

c. Adhesive tape has low peel strength too low

d. Difficulty cutting orifice with laser

Second air

bearing

prototype3

What was learnedPrototype




Origin of

idea1



First air bearing

prototype2




Second air

bearing

prototype3





Origin of

idea1



First air bearing

prototype2




Second air

bearing

prototype3


3.3 Motivation to design a HBP

The following reasons drove the decision to switch to a hydrostatic rather than aerostatic

bearing design for the proof of concept prototype:

• Incompressibility: instabilities, such as pneumatic hammer, can occur in aerostatic

bearings [1, 17, 18]. Hydrostatic bearings use incompressible fluids, and thus are

not generally susceptible to instabilities caused by compressibility.

• Viscosity: Commonly available oils can be on the order of 104 times [19] more

viscous than air. For this reason the bearing gap can be large while still providing

acceptable fluid resistance. Larger bearing gaps make the bearing’s performance

less susceptible to bore surface irregularities that result from manufacturing

errors. High viscosity makes the bearing less susceptible to the fluctuations in

performance that are caused by turbulence . The increased sensitivity is due to

the decreased Reynolds number of the flow.

• Surface compensation: Unlike aerostatic bearings, hydrostatic bearings use

viscous losses in the fluid as a means of restriction. The fluid can be restricted on

the bearing surface instead of through a small cross sectional area, e.g. an orifice

or capillary tube, which reduces the chances of clogging.

• Self-compensation: Self compensated bearings have higher load capacity and

stiffness than fixed restrictor bearings, as was mentioned in Chapter 1. Self-

compensated bearings require cross channeling between the pads and restrictors;

the formation of internal networks is an ideal demonstration of 3DWN

technology.

3.4 Satisfying precision requirements of the HBP

The bearing bore has to meet a certain level of precision in order to perform properly,

avoid contact with the shaft, and have a uniform fluid film. The proposed 3DWN

technology includes the use of off-the-shelf precision parts to meet the precision

requirements of the bearing. This section defines metrics for the off-the-shelf

components. The precision requirements for hydrostatic bearings are:

• Surface finish: The maximum peak-to-valley surface roughness of the hydrostatic

bearing components should not be greater than one-fourth of the bearing gap [1,

4].

• Bore diameter and roundness: The bore should not have errors great enough to

disturb the fluid flow or cause mechanical contact with the shaft. The diameter,

circularity, and surface finish errors should also not be greater than one fourth the

bearing gap.

3.4.1 Characterization of surfaces

The following information on surface finish is a summary from Applied Tribology,

Surfaces and their Measurement [20]. There are three important factors in defining the

roughness of a surface: 1) the roughness, which is a metric of the size of short-

wavelength irregularities, e.g. asperities, 2) the waviness, which is a metric of the long-

wavelength form error, and 3) the lay, which is the direction of the primary surface

irregularities. An example surface with these factors is displayed in Figure 3.3.

Lay

Total surface profile

Waviness profile

Roughness profile

Figure 3.3 Example surface roughness profile [20]

The average roughness, Ra, is the most common metric used to describe surface

roughness. This is defined, as shown in Figure 3.4, as the average deviation of individual

high and low points on the surface from the arithmetic mean height of the profile. In

order to incorporate the effect of waviness in Ra, the points are usually sampled over five

times the longest wavelength.

Ra

Figure 3.4 Determination of Ra value [21]

3.4.2 Satisfying surface finish requirements

One hypothesis associated with 3DWN technology is that the surface finish constraint of

the bearing is satisfied by the surface finish of the Template. It is generally good design

practice to specify that the max peak valley roughness, Ry, of the bearing bore should not

be greater than one fourth of the bearing gap height. Typically Ry is three times Ra [20].

If we accept this relationship, the metric for the required Template surface finish is

defined by Equation (3.1).

12oh

Ra ≤ (3.1)

3.4.3 Satisfying bore diameter and roundness requirements

Using replication to form the bearing bore is another hypothesis included in 3DWN

technology. As described in Chapter 1, the Template is wrapped around a mandrel to

inherit its precision diameter and roundness. Errors in the mandrel diameter may be

transferred to the 3DWN HBP. Equation (3.2) is used as a metric to ensure that the

bearing bore and mandrel surface asperities remain less than one fourth the gap height. In

Equation (3.2), Rε is the radius error, Cε is circularity error, and Ra is the surface finish.

43 o

CR

hRa ≤++ εε (3.2)

3.5 Design of 3DWN HBP bore surface features

The following functional requirements were outlined for the HBP bore surface features.

The successive subsections describe the design parameters used to satisfy the following

functional requirements.

1. Surface-self compensation: Compensation can be accomplished through

restrictors on the bearing surface instead of small-area restrictors that are

susceptible to clogging. Self-compensation requires cross-channeling between

pads and opposing restrictors, an ideal application to showcase 3DWN

technology.

2. Accommodation of overlap region on bore: There is an overlap region of the

Template on the bearing. The overlap has to be incorporated into the bearing

surface features without negatively affecting performance.

3. All features cut as 2D profiles: Conventional bearing features cannot be directly

applied to 3DWN bearings because all Template features have to be 2D through-

cuts.

4. Even number of pads: The bearing has to have an even number of pads as to so

each pad is directly opposed by another pad [6,16]. Odd pad arrangements can

lead to an imbalance of radial forces, which can move the shaft off center.

3.5.1 Inspiration for HBP surface feature design

The 3DWN HBP surface feature design was inspired by the self-compensated journal

bearing design shown in Figure 3.5 [22]. The bearing self-compensates through annular

restrictors that connect to opposed pads. The bearing in Figure 3.5 was used as a starting

point since its design is deterministic and therefore more easily realized. Modeling of the

3DWN HBP is included in Chapter 3.

A. Restrictor-pad cross-connection networks B. Self-compensated pad bore surface features

Figure 3.5 Hydrostatic self-compensated journal bearing [22]

The flow through the restrictor is responsible for the deterministic nature of the design

shown in Figure 3.5. Early surface self-compensated bearing designs relied upon bearing

flow that occurred in the same direction as the leakage flow. This is demonstrated in

Figure 3.6 A. The restrictor in Figure 3.6 A is not deterministic because leakage flow

affects the amount of flow to the pad. The models that capture the flow in Figure 3.6 B

are deterministic because the annulus is pressurized and the leakage flow is separated

from the flow to the pad.

Pin

Leakage

Flow

To Pad

Pin

Leakage

Flow

To Pad

Pin

To

Pad

Leakage

Flow

Pin

To

Pad

Leakage

Flow

A. Non-deterministic early restrictor design B. Deterministic restrictor

Figure 3.6 Annular restrictor designs for hydrostaic surface self-compensated bearings

3.5.2 HBP pad configuration

As mentioned in the beginning of this section, hydrostatic bearings require an even

number of pads that are equally circumferentially spaced around the bore. The pads need

to be in opposed pairs so as to balance the resultant forces, as shown in Figure 3.7 A. If

the pads are not opposed, as shown in Figure 3.7 B, uneven radial forces will cause the

shaft to move off center.

Bearing Pad

Force

Pocket Bearing Pad

Force

Pocket

Bearing Pad

Force

Pocket Bearing Pad

Force

Pocket

A. Even pad configuration B. Odd pad configuration

Figure 3.7 Comparison of pad configurations

The chosen layout of bearing surface features for the 3DWN HBP is shown in Figure 3.8.

A configuration of 2 axial sets of 4 pads was chosen since it simplifies the bearing design

by using the minimum number of pads that are required to fully constrain the shaft (4

degrees of freedom). The pads, not the restrictors, are located at the ends of the bearing to

further improve moment stiffness. The ratio of 2=D

L was chosen for compliance with

suggested shaft constraint configuration and St. Venant’s principle [1].

L

D

Pad Pocket

RestrictorDrain

Groove

Pad

Land

L

D

Pad Pocket

RestrictorDrain

Groove

Pad

Land

Figure 3.8 Chosen 3DWN HBP bore surface features layout

3.5.3 Overlap region of bearing bore

There is an overlap on the bearing bore, shown in Figure 3.9. As a result of the 3DWN

HBP is made from a wrapped sheet. At first glance the overlap region may look like a

glaring violation of required bore precision. In reality the overlap region can be used to

the advantage of the designer.

w

w

w

w

Overlap Region

Overlap Gap

Drain Pocket

w

w

w

w

Overlap Region

Overlap Gap

Drain Pocket

Figure 3.9 Geometric matching of overlap region

The overlap region creates an overlap gap on the bearing bore, as shown in Figure 3.9.

These gaps can be used to drain fluid from the pads. Fluid drains are desirable features in

hydrostatic bearings BECAUSE they prohibit flow interaction between the pads, making

the pad flow analysis deterministic. Geometrically opposed drain pockets can be cut into

the surface features if the arch length and position of the gap can be determined. The

pockets are geometrically opposed to balance the forces from the pads. The analytical

model describing the overlap region is included in Chapter 3.

The overlap region raises another issue of concern: the amount of Template required to

jump from one layer to the next is greater than the arch length of the overlap. As a result,

extra material needs to be added to each wrapped layer. A model for the extra material

required is included in Chapter 3. This model is expanded to map all 3D bearing and

internal features to features in the Template.

3.5.4 Restrictor design

There is an inherent problem with incorporating annular restrictors into 3DWN

technology: a true annulus cannot be created, as this would create a physical separation

between two parts of the restrictor. As a result, it would not be possible to fabricate the

entire bearing template in one piece. A solution to this problem is shown in Figure 3.10.

2D Cut Template

Annulus

Connection

Fluid

Inlet

No Flow

2D Cut Template

Annulus

Connection

Fluid

Inlet

No Flow

Figure 3.10 Single feed, double annulus restrictor configuration

The design in Figure 3.10 is a modified annulus that keeps all features of the design

physically attached to the rest of the Template (the cuts are represented by gray areas).

The design in Figure 3.10 has the following desirable features that distinguish it from

restrictors designed by Slocum [15,22,23] and make it ideal for 3DWN technology:

1. Annulus connection: The annulus is supported by a web of material connected to

the surrounding Template. If this connection is sized correctly, it doesn’t impede

the radial flow from the annulus to center hole by allowing the flow from ends of

the annuls to converge at the center hole, shown in Figure 3.10.

2. Single source feeding: Placing the restrictors back to back permit both to be fed

from a single source. This feature allows a single annular internal fluid network to

feed all the restrictors. Making the feed network annular is important to limiting

pressure variation on the bore, which can cause bore bulge. Bore bulge is

quantified in Chapter 3.

3.5.5 Full 3DWN HBP bore feature design

The 3DWN HBP bearing feature final design is shown in Figure 3.11. The axis of

symmetry in this figure corresponds to the center of the bearing, where an axial pad pair

meets. The pad bore surface features is repeated circumferentially 4 times around the

bore, creating a layout in the same configuration as shown in Figure 3.8.

s

rl

rl

l

l

l

l

m

Axis of Symmetry

m

Drain Pocket

Pad

Pocket

Restrictor

m

m

s

rl

rl

l

l

l

l

m

Axis of Symmetry

m

Drain Pocket

Pad

Pocket

Restrictor

m

m

Figure 3.11 3DWN self-compensated bearing pad bore surface features and geometric parameters

To simplify analysis of the bearing, the bore surface features in Figure 3.11 is designed to

be tunable with only one dimension: the pad width l. An analysis of bearing performance

is in included in Chapter 3. All other dimensions in Figure 3.11 were driven by

constraints and functional requirements placed on the bore surface features. An

explanation of the dimensions follows.

• Drain pocket width s: This dimension is driven by the overlap region arch length

shown in Figure 3.9. Each pad has to have the same size drainage pocket to

balance with the other pads

• Minimum feature length m: The minimum feature size that can be cut into the

template is dependent on the type of cutting process used, the thickness and kind

of Template material, the flow of the adhesive during wrapping and bonding, and

ensuring flow is fully developed over the restrictor. A model for analyzing the

effect this dimension has on bearing performance is included in Chapter 3.

Minimizing this dimension is desirable, as more area on the pad can be used by

the pocket, which increases the load capacity and stiffness.

• Leakage radius rl: The leakage radius rl is used to calculate the leakage flow from

the restrictor. To make this calculation deterministic, the restrictor is equally

spaced from the vertical and horizontal drain pockets. The dimension rl is driven

by m and s.

3.6 Design of Internal Channels

The following functional requirements were outlined for the HBP internal networks.

1. 2D through cuts: All the features for internal channels have to be 2D and extend

through the entire thickness of the Template.

2. Low fluid resistance in the channels: The fluid restriction caused by the internal

networks should be negligible (three orders of magnitude less) compared to the

restriction of the pads and restrictors.

3. Feed channels: An internal network is required to supply the restrictors with

pressurized fluid.

4. Cross-connection channels: Internal networks are required to connect the

restrictors with the pockets

5. Drain channels: Internal networks are required to remove fluid from the drain

channels.

Figure 3.12 shows the final design of the 3DWN HBP Template. Figure 3.12 shows the

Template cut into two sections since it is too long to be displayed as one piece. The

3DWN HBP has 13 wrapped layers, denoted by L1-L13 on the Template in Figure 3.12.

The successive subsections describe the design parameters used to satisfy the functional

requirements for the internal networks.

Bearing

topologyCross-connection channels

from restrictors to pads

Feed

channel

Drainage

channelsFluid

input

Fluid

output

L1 L2 L3 L4 L5 L6 L7

L8 L9 L10 L10 L11 L12 L13

Connection

holes

Bearing

topologyCross-connection channels

from restrictors to pads

Feed

channel

Drainage

channelsFluid

input

Fluid

output

L1 L2 L3 L4 L5 L6 L7

L8 L9 L10 L10 L11 L12 L13

Connection

holes

Figure 3.12 3DWN self-compensated bearing template

3.6.1 2D through-cut parameters

The same minimum dimension used in the bearing bore surface features of Figure 3.11,

m, was used as the internal channel width for all the HBP wrapped networks. Since the

internal networks are formed from thin sheets, the fluid resistance in the channels is

dominated by the channel height (in the radial direction), hc, as long as the relationship in

Equation (3.3) is true.

tnhm cc ⋅=> (3.3)

Where nc is the number of layers in the channel and t is the thickness of each layer.

3.6.2 Design of HBP fluid networks for low resistance

The primary source of fluid resistance in the bearing should be from the restrictors and

the pads. An analysis of channel resistance as a function of channel width, m, is presented

in Chapter 3. To insure the channel resistance as a function of channel height is negligible

compared to the bearing resistance (three orders of magnitude difference), hc be at least

10X the bearing gap height.

3.6.3 Feed channel design

The feed channel supplies the restrictors with fluid. Fluid enters the bearing at a single

source, labeled in Figure 3.12 as “Fluid input,” and is distributed circumferentially

around the bearing. The feed channel is subjected to the highest pressure within the

bearing, and can thus be the main cause of bore bulge. A model for bore bulge is

presented in Chapter 3. To limit the effect of bulging, the feed channel is designed to be a

symmetric annulus around the bearing circumference. Only one feed ring is required, for

each restrictor pair (Figure 3.10) is fed by a single fluid input.

3.6.4 Cross-connection channel design

The cross-connection channels in the 3DWN HBP connect the restrictors with the pads

opposed 180o around the bearing bore. In Figure 3.12 the channels are arranged in a “fish

gill” configuration in order to:

1. Minimize the distance the fluid has to travel between the restrictors and pads

2. Form a pattern that can be repeated through successive layers

3. Maximize the distance between the “Cross-connection channels” channels and the

“Connection holes” to reduce the possibility of leaks

3.6.5 Drainage channel design

The drainage channels collect fluid from the drain pockets and duct it to the outside of the

bearing. There are two channels, one for each set of 4 pads in the HBP bore surface

features. The drainage channels are located next to the feed channel to limit the amount

of material required for the Template.

3.7 Summary

This chapter describes the design decisions used in the 3DWN HBP and provides design

rules and metrics that can be applied to any 3DWN bearing design. The chapter begins

with the origins and conceptions of 3DWN technology. The remainder of the chapter

answers the following questions:

1. How can off-the-shelf parts be used to meet the precision requirements of fluid

film bearings? What level of precision is required from these parts?

2. How can conventional surface self-compensated bearing features to be cut as 2D

profiles and formed into a wrapped structure? How does the overlap region of the

wrapped Template affect the other features on the bearing surface?

3. How are internal wrapped fluid networks designed into a Template? How should

the channels be designed as to not affect bearing performance?

CHAPTER

4 MODELING AND ANALYSIS

This chapter describes the modeling used in the 3DWN HBP. The model used to describe

a wrapped structure is derived in the first section. The wrapping model is then converted

into a transformation algorithm to convert points from a 3D cylindrical structure to 2D

coordinates on the Template. The second part of the chapter describes the fluid flow and

bearing performance models. The chapter concludes with an analysis of the sensitivity of

manufacturing errors on bearing performance, as well as other sensitivities to be

considered when designing the 3DWN HBP.

4.1 Wrapping model 4.1.1 Describing a wrapped structure

When the Template is wrapped, it breaks contact with the mandrel in the overlap zone

between the first and second layer, as shown in Figure 4.1. The overlap zone is important

because l is longer than s, meaning the overall length of the layer will be longer than the

circumference of the mandrel.

l

xo

so

t

ro θo

θo

yo

F

Tangent

Mandrel

Template

Figure 4.1 Diagram of overlap region of layer one and two

The overlap region is modeled as a cantilevered beam, as shown in Figure 4.1. The base

of the beam has a zero slope condition and the end of the beam is tangent with the

mandrel. The assumptions made in the wrapping model are as follows:

1) Although 3DWN bearings are wrapped under tension, only the vertical

component of the tensile force is used in the analysis as stretching can be

considered negligible. This simplification is justified in a later section.

2) Wide beam (w > l) stiffening is also neglected as it does not affect the beam shape

for a fixed displacement [24].

l

xo

so

t

ro θo

θo

yo

F

Tangent

Mandrel

Template

lo

xo

so

t

ro θo

θo

yo

F

Tangent

Mandrel

Template

3) Bending stress does not exceed the yield stress (σys). Elastic deflection is insured

if Equation (4.1) is true.

ys

or

Etσ<

2 (4.1)

4) Changes in moment as a result of geometric changes are negligible with

deflection (ie lo ~ xo).

In Figure 4.1 there are two knowns (ro, t) and six unknowns (yo,F,lo,so,xo,θo). Geometric

conditions are used to write Equations (4.2)-(4.4) and bending beam theory for Equations

(4.5) and (4.6).

)cos1( ooo rty θ−+= (4.2)

ooo rx θsin= (4.3)

ooo rs θ= (4.4)

EI

Fly o

o3

3

= (4.5)

EI

Floo

2

2

=θ (4.6)

Where E is the modulus of elasticity and I is the moment of inertia.

Equation (3.6) is found by using Equation (4.7), which is taken from Bernoulli-Euler

beam theory.

dl

d

EI

M θ= (4.7)

Equation (4.7) is used to obtain the horizontal deflection. Using the geometry of a

differential beam element shown in Figure 4.2, Equation (4.8) can be written.

xo

l

dθdl

dy

y

x

dx

xo

l

dθdl

dy

y

x

dx

Figure 4.2 Model used for x-displacement of cantilevered beam

θcos

dxdl = (4.8)

Combining Equations (4.8) and (4.7), and substituting )( xlFM o −= for the moment in

the beam yields Equation (4.9).

θθ

cos)(

dx

d

EI

xlF o =−

(4.9)

Integrating over the intervals shown in Figure 4.1 yields Equation (4.10). Evaluating

Equation (4.10) results in a quadratic polynomial of xo with one possible root, shown in

Equation (4.11).

( ) ∫∫ =−oo

dF

EIdxxl

x

o

θ

θθ00

cos (4.10)

ooooF

EIllx θsin

22−−=

(4.11)

Combining Equations (4.2)-(4.6) and (4.11) results in Equation (4.12). Equation (3.12)

relates the known parameters of ro and t to the arc length of the overlap region, so. This

equation can be used for any layer of the wrapped structure by changing the variables

from first layer values (subscript o) to values of the local layer (subscript i). Values of s

have to be calculated numerically since Equation (4.12) cannot be simplified to a closed-

form solution.

o

oo

o

o

i

o

o

oo

o

o

r

sr

r

s

s

r

r

srt

s

rsinsin11cos1

2

30 −

−−

−+=

(4.12)

4.1.2 3D to 2D coordinate transformation

For Equation (4.12) to be useful, it has to be incorporated into an algorithm to transform

coordinates in a 3D cylindrical structure to 2D coordinates on the Template, as shown by

Figure 4.3.

Figure 4.3 Transformation of 3D cylindrical features to Template features

The first step in deriving a transformation function is to define the position of a feature in

the local layer in coordinates (r,θ,z), as seen in Figure 4.3. For simplicity, the derivation

will begin with the first layer (where r = ro), and then will be expanded to any layer.

Moving along the first layer, if the feature comes before the overlap region Equation

(4.22) describes its x position.

if oθπθ −≤ 2

θorx =

(4.13)

A percentage of lo has to be added to θor if the feature is within the overlap region. In

order to do this an expression for the position along the cantilevered beam in terms of θ

has to be derived. Figure 4.4 shows the local coordinate system for position along the

cantilevered beam.

Mandrel center

l’lo

θ’

θo

Figure 4.4 Local coordinate system for position along cantilevered beam

Equation (4.7) is used again to describe the deflection of the beam. Substituting the

moment with )( llFM o −= results in Equation (4.14).

dl

d

EI

llF o θ=

− )( (4.14)

Integrating over the range shown in Figure 4.4 results in Equation (4.15).

θ ′=

′−′

EI

lllF o

2

2

(4.15)

Equation (4.15) is related back to the coordinate system in Figure 4.3 by substituting for

2π - θ = θ’, resulting in Equation (4.16).

θπ −=

′−′

22

2

EI

lllF o

(4.16)

Combining Equation (4.16) with Equations (4.4) and (4.6) produces a polynomial with

one possible root that is shown in Equation (4.17).

( )

−−−=′

o

oo

s

rll

θπ211

(4.17)

The x coordinate within the overlap region, given by Equation (4.18), is found by

summing the length of material in contact with the mandrel and the difference between lo

and l’.

if oθπθ −> 2

( )

o

oooo

s

rlsrx

θππ

−−+−=

212

(4.18)

Combining Equations (4.2) and (4.4)-(4.6) creates an expression for lo, given by Equation

(4.19).

−+=

o

oooo

r

srrt

s

rl cos

2

3

0

0 (4.19)

Equation (4.19) can be used in any layer by changing the variables to values in that layer

(subscript change from 0 to n, where n is the number of the local layer). The local layer

number is found using Equation (4.20)

−=

t

rrn oint (4.20)

The radius in the local layer is expressed by Equation (4.21). This equation accounts for

the neutral axis of every layer being in the center of the Template thickness.

tnrr on

++=

2

1 (4.21)

Equation (4.19) can be used to find the overall length, Ln, of any layer in the wrapped

structure with Equation (4.22).

nn

n

nnn

n

nnnnn sr

r

srrt

s

rsrlL −+

++=−+= ππ 2cos

2

32

(4.22)

When equipped with Equations (4.13),(4.18), and (4.22) we may create an an algorithm

for transforming features from 3D to 2D. The algorithm is as follows:

1) Use Equation (4.23), which is equation (4.12) modified for any layer, to find sn.

n

nn

n

n

n

n

n

nn

n

n

r

sr

r

s

s

r

r

srt

s

rsinsin11cos1

2

30 −

−−

−+=

(4.23)

2) The x coordinate is found using Equation (4.24), which sums the length of the

interior layers and the distance to the feature in the local layer.

if 0=n

=x

θ

+ tro

2

1if

tr

s

o

o

2

12

+−≤ πθ

o

o

oos

tr

lstr

+−

−+−

+

2

1)2(

12

12

θππ if

tr

s

o

o

2

12

+−> πθ

if 0>n

∑−

=

+=1

0

n

i

iLx

θ

++ tnro

2

1if

tnr

s

o

o

++

−≤

2

12πθ

o

o

oos

tnr

lstnr

++−

−+−

++

2

1)2(

12

12

θπ

π if

tnr

s

o

o

++

−>

2

12πθ

(4.24)

3) The z coordinate simply becomes the y coordinate, shown in Equation (4.25).

zy = (4.25)

4.2 Modeling bearing performance 4.2.1 Fluid resistance modeling

The bore surface features and modeling used for the HBP is based on surface self-

compensated hydrostatic bearing theory from Slocum [1]. Figure 4.5 shows the fluid flow

of one restrictor-pad set. The HBP bore surface features is composed of 4 restrictor-pad

sets placed back to back (restrictors fed by same source), arranged π/2 radians apart

around the bore, as was shown in Chapter 2.

a

b

Fluid flow

φ

l

rp

Leakage flowPad flow

Restrictor flow

rl

rh

rarr

a

b

Fluid flow

φ

l

rp

Leakage flowPad flow

Restrictor flow

rl

rh

rarr

Figure 4.5 Flow over bearing bore surface features

To review from Chapter 1, the bearing self-compensates because each restrictor is

connected to the pad located π radians around the bore. When the shaft is displaced

downward the restrictor resistance changes such that pressure is increased under the shaft

and decreased above the shaft. The same fluid circuit shown in Chapter 1, shown again in

Figure 4.1, is used to model the HBP.

Figure 4.6 Fluid circuit of one set of opposed pads in the HBP

Ps

RUR

RLPRLR

RUP

RLL

RUL

PA

Ps

RUR

RLPRLR

RUP

RLL

RUL

PA

Ps

RUR

RLPRLR

RUP

RLL

RUL

PA

Ps

RUR

RLPRLR

RUP

RLL

RUL

PA

The flow through the bearing is modeled as fully-developed, viscous flow. Using the

same method of evaluating resistance presented in Chapter 1, The Navier-Stokes

equations reduce such that the nominal resistances are as follows. The derivation of these

resistances can be found in any common fluids textbook [25].

Equation (4.26) for the upper restrictor resistance, RUR, and lower restrictor resistance,

RLR.

3

ln6

h

r

r

RRh

a

LRURπ

µ

==

(4.26)

Equation (4.27) for the upper pad resistance, RUP, and lower pad resistance, RLP.

( )

++

+==

p

p

LPUP

r

lr

h

l

hbaRR

ln66

133

µ

π

µ

(4.27)

Equation (4.28) for the upper leakage resistance RUL, , and lower leakage resistance, RLL.

)2(

2ln6

3 ϕπ

π

π

µ

−

==h

r

r

RRr

l

LRUR

(4.28)

The factor )2(

2

ϕπ

π

− in Equation (4.28) accounts for the no-flow zone. This zone occurs

since there is no pressure differential to drive the flow between restrictors.

4.2.2 Resistance ratio

The resistance ratio between the restrictors and pads is defined by Equation (4.29)

UP

LR

LP

UR

R

R

R

R==γ (4.29)

Slocum [1] suggests the resistance ratio should be between 3 and 4 for uniform stiffness.

The resistance ratio chosen for the HBP was 3.25. This ratio was met by tuning the

dimension l on the bearing pad.

4.2.3 Derivation of effective pad area

The pressure profile over each pad is trapezoidal in shape, as shown in Figure 4.7.

Restrictor

a+2rp

ll

b+2rp

Pp

Restrictor

a+2rp

ll

b+2rp

Pp

Zone 2

Zone 3

Zone 1

Figure 4.7 Pressure profile over bearing pad

The purpose of deriving an effective area is so the resultant force from the pad can be in

the form of Equation (4.30).

effp APF ⋅= (4.30)

Where Pp is the pressure in the bearing pocket and Aeff is the effective area. The total

effective area is found by adding up the volume of each zone in Figure 4.7 and factoring

out the pressure.

The vertical force contribution of Zone 1 is determined by integrating the trapezoidal

force element around the arc length of the pad, which is evaluated in Equation (4.31)

( ) ( )

+++=++= ∫

+

r

rarlrbPdrlrbPF

p

ppr

ra

ppZo

p

2

2sin22cos22 0

2

2

001 θθ

(4.31)

The vertical force contribution of both Zone 2 triangles is determined using Equation

(4.32) by multiplying their total volumes by

+

r

ra p

2

2sin .

++=

r

rarblPF

p

ppZ2

2sin)2(2

(4.32)

The vertical force of the four Zone 3 pyramidal sections is determined using Equation

(4.33) by summing the volume of the sections and multiplying by

+

r

ra p

2

2sin .

+=

r

ralPF

p

pZ2

2sin

3

4 2

3 (4.33)

Summing Equations (4.31)-(4.33) and factoring out Pp gives the total effective area over

which each pad acts, shown in Equation (4.34).

( ) ( )

+

+++++=

o

p

popeffr

rallrbrlrbA

2

2sin

3

4222

2

(4.34)

4.2.4 Derivation of bearing stiffness

To develop a stiffness model the bearing is oriented such that four pads are vertical and

four are horizontal, as shown in Figure 4.1. Only the vertically oriented pads are loaded;

shaft deflections are small enough to assume the horizontal pads contribute zero stiffness

in the vertical direction.

y

xz

Shaft

Bearing

Pad stiffness

Load

y

xz

Shaft

Bearing

Pad stiffness

Load

Figure 4.8 Pad stiffness configuration of HBP (gap greatly exaggerated)

The voltage divider rule (P=QR for fluid circuits) can be applied to find the pressure

difference between the pads by using the circuit in Figure 4.6 and Equations (4.26) and

(4.27). Each resistance in the circuit is a function of the gap height and will change with

displacement of the shaft. Multiplying the pressure difference between the upper and

lower pads by the effective area gives the resultant force on the shaft. Equation (4.35)

gives the force as a function of shaft displacement, δ.

++

−

+−

−+

+

−=

33

3

33

3

)(

1

)(

)(

1

)(

1

)(

)(

1

δδ

γ

δ

δδ

γ

δ

oo

o

oo

o

seff

hh

h

hh

hPAF

(4.35)

The stiffness for one set of opposed pads, given by Equation (4.36), is the derivative of

Equation (4.35) in terms of displacement.

( ) ( )

++

−−

+

++

−+

+−

−=

33

42

33

3

44

)(

1

)(

1

)(

1

)(

)(

1

)(3

δδ

γδ

δδ

γδ

δδ

γ

oooo

ooseff

hhh

hhh

hhPAk

( ) ( )

++

−−

+

++

−−

+−

−−

33

42

33

3

44

)()(

1

1

)()(

1

)()(

1

δ

γ

δδ

δ

γ

δδ

δ

γ

δ

oooo

oo

hhh

hhh

hh

(4.36)

The total stiffness of the HBP is obtained by multiply Equation (4.36) multiplied by a

factor of 2.

4.3 Sensitivity Analysis

4.3.1 Justification for using non-precision cutting processes

In Figure 4.9 the subscript (o) denotes ideal values and the curves are generated from

Equation (4.36).

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

0.6

0.8

1

1.2

1.4

Expected εl – from

through-cutting

Expected εh – from mandrel

( )

l/lo, h/ho

k/ko

k(l)

k(h)(

Figure 4.9 Sensitivity of the HBP stiffness to manufacturing errors for h/R = .002, h/l = 0.011

The principle errors that affect bearing stiffness (k) are:

1) Errors in the fluid film height (εh), caused by manufacturing errors in the mandrel

or the shaft used in the bearing

2) errors in the dimensions of the bore surface features (εl), caused by errors in the

through-cutting process used for the Template.

Tolerances of the mandrel radius are typically much better (10X) than the tolerances of

the Template. The stiffness is much more sensitive to errors in the mandrels since

hl

hl εε<< and

hk

1∝ , whereas lk ∝ . This means a non-precise through-cutting process

(like laser or waterjet) can be used to manufacture the Template, as the performance of

the bearing will be dependent on the precision of the off-the-shelf parts and not the

Template.

4.3.2 Sensitivity of performance to internal channel errors

Internal channel resistance, Rc, being significant compared to the resistance of the bearing

features, Rb, can cause a negative effect on bearing performance. Errors in Template

thickness and mandrel radius cause misalignment of internal features, as shown in Figure

4.10, which decrease the effective channel width and can choke bearing fluid flow.

wo

w

Layer n+1

Layer n

Flow

wo

w

Layer n+1

Layer n

Flow

Figure 4.10 Channel constriction as a result of internal feature misalignment

Using Equation (4.24), values of m were calculated for different ideal widths (mo) by

varying mandrel radius (r) and the Template thickness (t) over the expected error range of

the off-the-shelf parts used for the HBP. The comparison between resistances was made

using the approximations in Equations (4.37) and (4.38). The results of this analysis are

displayed in Figure 4.11.

3

1~

o

bh

R (4.37)

3

1~

mRc (4.38)

0.9

99

75

0.9

99

85

0.9

99

95

1.0

00

05

1.0

00

15

0.8

00

00

0.8

88

89

0.9

77

78

1.0

66

67

1.1

55

56

7.94E-09

7.96E-09

7.98E-09

8E-09

8.02E-09

8.04E-09

8.06E-09

8.08E-09

8.1E-09

8.12E-09

r/r o t/t o

m o /h o = 500

R c /R b

0.9

99

75

0.9

998

5

0.9

999

5

1.0

00

05

1.0

00

15

0.8

00

00

0.8

88

89

0.9

777

8

1.0

66

67

1.1

55

56

0.0000074

0.0000076

0.0000078

0.000008

0.0000082

0.0000084

0.0000086

0.0000088

0.000009

0.0000092

r/r o t/t o

m o /h o = 50

R c /R b

0.9

99

75

0.9

99

85

0.9

99

95

1.0

00

05

1.0

00

15

0.8

00

00

0.8

88

89

0.9

77

78

1.0

66

67

1.1

55

56

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

r/r o t/t o

m o /h o = 5

R c /R b

Figure 4.11 Internal channel resistance sensitivity to expected error range of t and R for t/R = 0.01

The trends in Figure 4.11 were used to choose a channel ratio of mo/ho = 50 for the HBP,

which corresponds to mo = 2.54mm. This value of mo insures manufacturing errors will

not affect bearing performance and also gives a buffer for glue squeeze into the channels.

It is important to note that angular misalignment (e.g the Template not being

perpendicular to the mandrel, resulting in spiraling error during rolling) was not

considered in the sensitivity analysis. This decision was made because angular

misalignment is an error that could be actively corrected for during the rolling process,

whereas manufacturing errors are inherent to the precision parts.

4.3.3 Justification for neglecting tension in the wrapping model

As was mentioned in the beginning of this chapter, stretching effects due to tension were

neglected from the wrapping model. This assumption was validated in two ways:

1) The strain over the entire Template was calculated to be on the same order as the

manufacturing errors used to make the through-cuts.

2) An FEA analysis was conducted on the section of Template in the overlap region

to determine if tension would change the beam shape and resultant length, as

shown in Figure 4.12. The maximum calculated change in length over the

cantilevered beam was 0.01%, which can be regarded as negligible.

yδ

T

εδ

yδ

T

εδ

Figure 4.12 Error caused by wrapping tension

4.3.4 Sensitivity to bore bulge to channel placement

Early 3DWN prototypes had the problem of bore bulging during pressurization of the

bearing, As was discussed in Chapter 2. Bore bulge, seen in Figure 4.13, occurs when

internal fluid feed channels from pressurization, reducing the diameter of the bore.

Pressure

ChannelBulge

Pressure

ChannelBulge

Figure 4.13 Bore bulge resulting from pressurized channels

An FEA sensitivity analysis was performed to see how internal channel dimensions affect

bore bulge in order to determine the minimum amount of material required between the

bore and the feed channels. Using an m/t ratio of 20 and bronze shim stock,

corresponding to one layer of the Template, the bore bulge was three orders of magnitude

smaller than the bearing gap, making its affect on bearing performance insignificant.

4.3.5 Appropriate wrapping tension to compress Template deformities

Deformities in the Template were observed preventing contact between the layers in early

prototypes. To compress such deformities an appropriate amount of wrapping tension had

to be determined. The Template thickness was determined by using a level of constant

tension that could easily be produced in a lab. The tension used in wrapping was 90 N (20

lbs), which was produced by four 22.25 N (5 lbs) barbell plates. The satisfactory amount

of vertical deflection of one Template layer was chosen to be 254 µm, a value which is on

the same order as the thickness. Further detail on the wrapping process is described

Chapter 4.

The model used for the FEA tension analysis is shown in Figure 4.14. The pressure

applied to the top surface of the beam in Figure 4.14 was determined by using a thin-wall

pressure vessel model. If adjacent layers are permitted to slide against one another, which

liquid-state glue permits, then the tension pulling on π radians of the Template will be

reacted by a pressure analogous to that on half of a cylindrical pressure vessel.

Pressure

T T

δy

Template

Pressure

T T

δy

Template

Figure 4.14 Deflection of Template due to tension

FEA results from the loading condition in Figure 4.14 are shown in Figure 4.15. All

analyzed thicknesses meet the desired deflection. The Template thickness was chosen to

be 127 µm, which corresponds to the thinnest bronze shim stock commonly available, to

obtain the maximum possible deflection.

0

1000

2000

3000

4000

5000

6000

254 203.2 177.8 127 101.6

t (microns)

δy

(mic

rons)

Figure 4.15 FEA Results from deformity deflection under tension

4.4 Summary This chapter the modeling and analysis used in designing the 3DWN HBP. A model to

transform coordinates in a 3D structure to a 2D Template is presented. Conventional self-

compensated hydrostatic bearing theory is used to model the HBP performance. The

sensitivity of bearing performance to manufacturing errors is explored. The sensitivity

analysis concludes that bearing stiffness sensitivity is dominated by defects in the

precision ground mandrel and not errors in the cutting process. The chapter concludes

with verifications of simplifications made in the wrapping model and calculations of

design and manufacturing parameters that ensure proper bearing performance.

CHAPTER

5 MANUFACTURING A 3DWN HBP

This chapter describes the methods used in manufacturing a 3DWN HBP. The chapter

begins with a review of failed strategies of adhering wrapped layers together. The

remainder of the chapter describes the manufacturing processes used in making the final

HBP. The manufacturing processes are broken into three sections: 1) how the Template

was cut, 2) how the Template was wrapped, and 3) how the Template was packaged into

a housing.

5.1 Failed attempts at adhering wrapped layers Table 5.1 summarizes the methods attempted and the causes of failure in trying to adhere

3DWN wrapped layers. A concern in early HBPs was that a liquid adhesive would clog

the internal channels. Another concern was adhesive strength; early 3DWN aerostatic

prototypes were made with double-stick tape, which was not strong enough to prevent

bore bulge. The methods of adhesion stated in Table 5.1 were all chosen for high bond

strength and minimized flow into the internal channels.

Table 5.1 Failed attempts at adhering wrapped layers

Type of adhesive Why used/failure

Template

Epoxy

tape

3M heat set epoxy tape [26]

Why used: Conveniently in roll form. Rolled onto Template Cause of failure: Adhesive cures to consistency of boot leather. Creeping was observed, which warped the bore

Template

w/ solder

Mandrel

Solder

Why used: In bench-level tests, solder wicked between layers. Solder also can adhere bronze, which is commonly used in bearings Cause of failure: Surfaces would tin, but adjacent layers did not adhere because removing flux proved impossible.

Adhesive

film

Template

3M heat set adhesive film [27]

Why used: Aircraft grade, Ultra high strength, stiff adhesive. Described as “low flow.” Cause of failure: Adhesive clogged channels because could not control flow. Adhesive also squeezed out between layers, which resulted in bore expansion and warping.

5.2 Template fabrication

5.2.1 Waterjet cutting

The 3DWN HBP was fabricated with an abrasive waterjet cutter, as shown in Figure 5.1.

The Template was clamped between two sheets of protective plastic to protect the

Template surface finish and keep it secured to the bed. The Waterjet process was used

because the Template is made from 510 phosphor bronze. It is undesireable to laser cut

this material as it would release poisonous fumes.

Protective

plastic

Topology

Coupling

Protective plastic

Template

Coupling

Template

A. Before cutting B. After cutting

Figure 5.1 Template being cut in waterjet

5.2.2 Fixturing of Template within waterjet

The Template for the HBP had to be cut in segments because it was longer than the

waterjet bed. A kinematic fixturing system, shown in Figure 5.2, was incorporated into

the Template to maintain alignment between segments. Each coupling feature has a

single point contact with a pin on the 3-pin plate and is pre-loaded with a flexure. The 3-

pin-plate was cut in the same waterjet as the Template and positioned at the same zero to

preserve the alignment of the machine axes.

Clamping plateTemplate

Coupling feature

Perforation

Preload

Flexure3-pin plate

Clamping plateTemplate

Coupling feature

Perforation

Preload

Flexure3-pin plate

Figure 5.2 Kinematic fixture for waterjet cutting, waterjet cutting setup

The sides of the Template, which include the coupling features, are not necessary for

bearing function and are designed to be easily removed after fabrication. The perforations

incorporated into the Template, shown in Figure 5.2, allow the coupling features to be

snapped off.

5.3 Template wrapping 5.3.1 Rolling jig

The rolling jig shown in Figure 5.3 was used to wrap the HBP. All of the components in

the jig were made from Delrin and cut on a waterjet. Delrin was the chosen because it has

a low coefficient of friction against steel, allowing the mandrel to turn easily. Delrin is

also commonly used for flexures, making it a good choice for the friction break.

Mandrel

Crank

Bearing blocks

Friction clamp

Constraining clamp

Figure 5.3 3DWN bearing rolling jig

The rolling jig was mounted to an optical table. The constraining clamps constrain the

mandrel from moving axially, and allow the crank to be positioned so it clears the table.

The friction break allows tension to be maintained on the Template during pauses in

wrapping.

5.3.2 Alignment of Template to mandrel

Figure 5.4 shows how the Template was mounted to the rolling jig, and how the rolling

jig was mounted to an optical table. Before the HBP Template was mounted all burrs

were removed. Surfaces to accept adhesive were prepared by light sanding and cleaning

with alcohol.

Weights

Tension

distributor

Template

Level

Mandrel

Rolling crank

Tension

balancer

Figure 5.4 Template mounted on rolling jig

The Template was aligned on the rolling jig by using the following process:

1) The sacrificial tabs at the beginning of the Template were affixed to the mandrel

with hose clamps.

2) Weights were hung at the end of the Template to provide the calculated required

wrapping tension.

3) The hose clamps were tapped outwards to pull the leading edge of the Template

flush to the mandrel.

4) The sine error was measured over the horizontal Template length and adjusting

with the Tension distributor.

5) The hose clamps were slowly loosened until the Template was level.

The use of a level to measure tension over the Template was realized after observing the

relationship in Equation (5.1).

dw

dT=α (5.1)

Where α is the roll angle across the width of the Template, w is the template width, and T

is the applied tension.

5.3.3 Adhesion of wrapped layers

Superglue gel was used to adhere adjacent wrapped layers within the 3DWN HBP. Gel

over liquid super glue was chosen because it doesn’t cure as quickly and doesn’t wick.

The mandrel was coated with a light oil to prohibit the superglue from adhering. The

Template surface was cleaned before the glue was applied using alcohol to remove any

oil, as shown in Figure 5.5A.

Hose clamp

Alcohol

Template

Mandrel

Hose clamp

Super glue

Template

Mandrel

A. Cleaning surface with alcohol B. Applying superglue gel

Figure 5.5 Adhering adjacent layers within the HBP

Adhering all surfaces in the first layer is important because the bore surface features have

to conform to the mandrel. Figure 5.5B shows how the superglue was applied to the first

layer. In the first layer, excess glue that spilled into the bearing features was removed

after the mandrel was taken out of the bearing. Glue within the channels was removed by

dabbing it out through the cutouts in the second layer. Glue was applied to layers beyond

the first in such a way to limit spreading into the channels. As a result, closely located

features in the Template could not be sealed from one another.

5.4 Packaging the Template in a housing 5.4.1 Joining Template and housing

A housing is required for the wrapped Template because it provides a connection to a

pressurized fluid source and rigidity for mounting. The first step in packaging the

Template was to affix it in the center of the housing. The Template was glued to two

concentric centering shims, as shown in Figure 5.6. The shims were then glued to the

housing.

Aluminum sleeve

Wrapped

Template

Centering shims

Aluminum sleeve

Wrapped

Template

Centering shims

Figure 5.6 Centering of Template within housing

5.4.2 Prepping the HBP for casting

The wrapped Template has openings for fluid input and output. The methods used in

preventing casting epoxy from entering these openings are shown in Figure 5.7. The

inlets and outlets on the Template were covered with electrical tape, shown in Figure

5.7A. Then the openings in the housing were plugged with grease packed foam, shown in

Figure 5.7B. The plugs, when preloaded against the outer surface of the wrapped

Template, prohibit epoxy from building up over the electrical tape and filling the

openings in the housing.

Epoxy

sealMandrel

Aluminum sleeve

Inlet

covered with tape

Grease plug

Aluminum

sleeve

A. Sealing Template from epoxy B. Grease plug for blocking epoxy

Figure 5.7 Preparations for casting

A seal made of latex was attached to the wrapped Template and the housing. Figure 5.7A

shows the seal before it was is attached to the housing.

5.4.3 Casting the Template into the housing

The epoxy used to cast the Template into the housing was 3M Scotchcast [28]. The epoxy

was funneled into the gap between the wrapped Template and housing and allowed to fill

to the top of the housing, as shown in Figure 5.8. The epoxy was left to cure for 24 hours

before the next packaging process.

Mandrel

Epoxy

Aluminum sleeve

Figure 5.8 Wrapped Template cast in housing

5.4.4 Finishing procedures

The grease plugs were removed after the epoxy dried. Notice in Figure 5.9 how the

drainage plugs in the wrapped template are visible; the grease plugs and electrical tape

prevented epoxy from reaching the drainage ports in the Template.

Drainage ports after cleaning

Aluminum sleeve

Figure 5.9 Drainage ports with grease plugs removed

The mandrel was removed with an arbor press. The final process in packaging the HBP

was to remove the excess superglue in the bore surface features, shown in Figure 5.10.

The glue was carefully cut out of the bore using a hobby knife.

Template

topology

Residual super glue

Figure 5.10 Residual super glue to be removed from bearing bore surface features

5.5 Summary This chapter explains how the 3DWN HBP was manufactured. Unsuccessful attempts

adhering the wrapped layers are presented in the first section. The remainder of the

chapter describes in three sections the manufacturing processes used for the final HBP.

The first section describes how the Template was cut in segments using a waterjet. The

second section describes how the Template was wrapped and adhered. The third section

describes how the bearing was finished and packaged into a housing.

CHAPTER

6 EXPERIMENTAL VERIFICATION

The purpose of this chapter is to verify the feasibility of 3DWN technology by comparing

the HBP performance to the models developed in Chapter 3 and determining if the bore

precision is within the metrics outlined in Chapter 2. This chapter begins with a

description of the experimental setups used to measure bearing stiffness and bore

geometry. Results from these tests are presented and compared against theoretically

predicted and ideal values. A discussion and interpretation of the results and sources of

error is included. The financial benefits of 3DWN technology are exhibited in a cost

comparison between the HBP and a babbitted, self-compensated bearing of the same size

and surface feature layout.

6.1 Experimental setup 6.5.1 Parameters of the 3DWN HBP used in experimentation

The parameters of the 3DWN HBP used for experimentation are shown in Table 6.1. The

methods behind generating and outlining the parameters in this bearing are described in

Chapters 2-4.

Table 6.1 Parameters of the 3DWN HBP used in experimention

Values for HBP

experimentation

Dimensionsgap height h (µm) 50.80

pad pocket width a (mm) 23.72

pad pocket length b (mm) 10.71

pad land width l (mm) 4.77

pad coner radius r p (mm) 1.27

radius of restrictor hole r h (mm) 1.27

radius of restrictor anulus r a (mm) 3.81

radius of restrictor feed channel r r (mm) 6.35

radius of leakage flow field r l (m) 17.86

superglue thickness (m) 25.40

Oil SupplySource pressure Ps (kPa) 606.7

SAE 40W Oil viscosity µ (N*s/m^2) 0.119

Material propertiesTemplate material 510 phosphor bronze

Housing material 6061-T6 aluminum

Epoxy 3M-Scotchcast

Calculated performance propertiesResistance ratio γ 3.23

Stiffness k (N/µm) 25.77

Precision of off-the-shelf parts

Mandrel radius r o (mm) 50.9016 +/-0.0127

Shimstock surface finish (µm) Ra = 0.5

Shim stock thickness t (mm) 0.127 +/-0.0127

a

b

l

rl

rp

rr

ra

rha

b

l

rl

rp

rr

ra

rh

5.1.2 Experimental setup for stiffness testing

The setup in Figure 6.1 was used to measure the stiffness of the HBP. The bearing was

loaded by hanging weights, equal distance from bearing center, on the shaft. The loading

varied form 0 to approximately 70N. A load of 70N is not enough to deflect the shaft

over the bearing’s total operating range (ho/2) [1], but is enough to measure the stiffness

because, from the stiffness model, the deflection should remain linear with R2 ~ 0.9999X.

Two capacitance probes with 40nm accuracy were used to measure displacement. 100

data points over 10 seconds were collected at each load level to average out vibration.

Each capacitance probe was mounted directly to the bearing as to eliminate parasitic error

from deflection of the test rig. Oil runoff was collected in a pan and recycled back

through the system.

Test rig

Oilsupply

Metrologyframe

Cap. probe

Shaft

Oil reclaim

3DWNHBP

Test rig

Oilsupply

Metrologyframe

Cap. probe

Shaft

Oil reclaim

3DWNHBP

Figure 6.1 Experimental setup for testing stiffness

The oil pressurization system is shown in Figure 6.1. Oil was pressurized by connecting

shop air at 690kPa to the top of the oil reservoir and letting oil flow out the bottom. The

oil was then forced through a 0.64 µm filter. Source pressure was measured after the

filter. Compressed air was used because 1) of availability of air canisters in the lab 2) the

bearing could be run at a safe low pressure, and 3) it required no extra pumping

equipment. High viscosity oil (SAE 40W) was chosen to make the bearing less

susceptible to manufacturing errors, as it allowed the Reynolds number to remain low for

a relatively large (midrange for hydrostatic bearings [1]) bearing gap. The bearing was

less susceptible to manufacturing errors by having a large gap.

Ps

gauge

0.64 µm

Filter

Test rig

Input air

Oil

reservoir

Figure 6.2 Oil pressurization device

5.1.3 Bore measurement

Bore errors were tested in a CMM machine with 5µm accuracy. Each end of the bearing

(front and back) was measured 3mm below the edge of the bore, as seen in Figure 6.3A.

In each test 100 points were taken around 300o of the bore; 60o were omitted to avoid the

overlap zone, shown by Figure 6.3B. The CMM used was not capable of outputting

arrays of measurement coordinates. Test data was instead reported using error

distribution parameters mean, standard deviation, and min and max measured error.

CMM tip

Template

topology

Clamp

Bearing

front

Bearing rear

5π/3 rad bore

measurement

Overlap

A. CMM measurement setup B. Measurement range around bore

Figure 6.3 Bore precision testing using a CMM

5.2 Stiffness results and discussion 5.2.1 Stiffness test results

The data displayed in Figure 6.4 is the mean displacement of the shaft, derived from

measurements of both capacitance probes. The data from each probe was combined for

ease of graphic representation and because the purpose of the experiment was to measure

the overall bearing stiffness.

y = 0.0383x + 0.1613

R2 = 0.99

y = 0.0198x - 0.0159

R2 = 0.9951

y = 0.0388x - 0.0002

R2 = 1

y = 0.0382x - 0.0159

R2 = 0.9951

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70

Force (N)

Dis

pla

ce

me

nt

(mic

ron

s)

Measured data

Theory

Asperity stiffness

Measured, asperitystiffness removed

Figure 6.4 Measured stiffness vs. theory

The initial measured data indicated the bearing was twice as stiff as theory predicted, As

can be seen from Figure 6.4. When the bearing was turned on, the shaft was observed to

deflect down (in the direction of gravity) by approximately 1µm. This indicated that the

shaft was making contact with the bearing bore.

5.2.2 Sources of error in stiffness data

The stiffness of the oil film in Figure 6.4 was obtained by turning off the oil, measuring

the contact stiffness, and removing it from the measured data. The measured data with

asperity stiffness removed is within 1.6% error of the theory.

The contact stiffness and the oil film can be added as springs in parallel if the shaft was

touching the bore at only a few points and not as a Hertzian contact, allowing a full oil

film to develop. This hypothesis was validated by calculating the Hertzian contact

stiffness, which is found to be 48 times greater than the measured contact stiffness.

The downward shaft deflection at startup can be explained by leaks in the internal

networks. Visible leaks were apparent between the pressure source and the restrictor-pad

networks, shown in Figure 6.5 (normally when the shaft is removed there should be no

fluid flowing from the centers of the restrictors). Additional leaks were observed between

layers at the bearing pads. Leaks were caused by lack of adhesion between the layer.

Leakage flow

Primary flow

Restrictors

Leakage flow

Primary flow

Restrictors

Figure 6.5 View of leakage flow (shaft removed from bore)

The effects of leakage flow can be inspected by incorporating the resistance of upper and

lower restrictor leaks ,RURL, RLRL, and the upper and lower pad leaks, RUPL, RLPL, into the

fluid circuit shown in Figure 6.6,

Ps

RUR

RLPRLR

RUP

RLRL

RURL

PA

RUPL

RLPL

Ps

RUR

RLPRLR

RUP

RLRL

RURL

PA

RUPL

RLPL

Figure 6.6 Bearing fluid circuit including leakage flow

Figure 6.7 shows the ideal theoretical stiffness curve with the stiffness curve generated

using the circuit in Figure 6.6.

y = 0.0393x - 0.0289

R2 = 1

y = 0.0393x + 0.0289

R2 = 1

y = 0.0389x + 8.2732

R2 = 0.9995

y = 0.0529x + 11.345

R2 = 0.9967

-15

-10

-5

0

5

10

15

-500 -400 -300 -200 -100 0 100 200 300

Force (N)

Dis

pla

ce

men

t (m

icro

ns

)

Ideal, (+) Disp

Ideal, (-) Disp

Leak, (+) Disp

Leak, (-) Disp

Figure 6.7 Theoretical stiffness with and without leaks

Figure 6.7 shows that a large shift from the neutral position is possible without

significantly changing the bearing stiffness (positive displacement in the direction of

gravity). The leakage gap heights used in generating Figure 6.7 were 101.6µm for the

internal channels at the lower restrictor and 25.4µm for the upper bearing pads. Both of

these values are within the manufacturing tolerance range of the Template. It is important

to note that these leakage values are most likely on the same order as what could be

expected, and are used to demonstrate possible trends that could result from leakage.

Combined effects from the horizontal pads, which could cause further off-center

displacement of the shaft are not included in this analysis.

An attempt was made to repair the leakage in the pads. During repair the bearing was

dropped, which warped the bore. Further tests were impossible given that the HBP was

damaged. Also, bearing performance was observed to degrade quickly. The bearing

became non-repeatable and more compliant with successive tests during testing in the

reversed configuration. Increasing compliance would result from increased leakage,

indicating there was adhesive failure between wrapped layers. All data taken after the

first trials was not considered as a result of the mentioned errors.

5.3 Results from bore measurements

Table 6.2 contains the CMM measurement results of the bearing bore. Both ends of the

bearing are within 0.03% of the ideal mean diameter (µD). The front and rear of the

bearing are within 5σ and 2σ, respectively, of the tolerable error range. Although the

precision of the bore is not ideal, it is encouraging that the front of the bearing is 1σ away

from spec, and both ends are within 15.2µm of the desired mean diameter.

Table 6.2 Measurements of bore at front and rear of bearing

Measurement Ideal Bearing front Bearing rear

µ D (mm) 50.902 50.897 50.917

σ(µm) 8.64 7.62 12.70

Max εD(µm) 25.40 12.70 43.18

Min εD(µm) -25.40 -35.56 -20.32

The HBP measured was the first bearing to be adhered with superglue, which made the

wrapping process difficult. The data in Table 6.2 is promising, and with further

refinement and practice of the wrapping and adhesion process the bore will ideally be

made to spec.

5.4 Cost comparison

The HBP was compared to a babbitted bearing of similar size and bore surface features

[11] to demonstrate the savings possible with 3DWN technology. Table 6.3 shows a 10X

reduction in cost with 3DWN. Further savings might be possible once a standard 3DWN

manufacturing process is established.

Table 6.3 Cost comparison 3DWN and conventional bearing

Bearing type3DWN Self-Compensated

Hydrostatic

Babbitted Self-

Compensated

Materials $50.00 $30.00

Topology Machining $180.00 $700.00

Housing/Channels Machining $30.00 $5,000.00

Assembly Labor $320.00 $0.00

Total Cost $580.00 $5,730.00

5.5 Summary Although contact between the shaft and bore was observed, the fluid film stiffness

matched theory within 1.6% error after accounting for the contact stiffness. Measuring

the contact stiffness to be much lower than the calculated Hertzian stiffness indicates the

contact occurred only at a few points, which would have still allowed a fluid film to

develop between the bore and shaft.

The mean bore diameter was measured to be within 0.03% error of the mandrel with

errors that lie within 5σ of the tolerable error range in the front of the bearing and 2σ in

the rear. Measurements within 5σ are encouraging. Ideally with more refinement 6σ can

be obtained. As a result of the shimstock used, Table 6.1 shows that the surface finish of

the Template is within the metric defined in Chapter 2.

The most compelling result of this chapter is the cost comparison between 3DWN and

babbitted bearings. Savings could most likely be further increased with improvements

and practice in the wrapping process.

CHAPTER

7 SUMMARY

This chapter outlines the scholarly contributions, impact, and future work associated with

this research.

7.1 Scholarly Contributions The following contributions are a result of the work presented in this thesis:

7. A new approach of fabricating low-cost and flexible hysdrostatic bearings –

3DWN technology is a new method of making a precision bearing bore. Surface

replication by wrapping is a deviation from conventional bearing manufacturing

practices; all current fluid film journal bearings require at least one precision

process in their construction.

8. A method of decoupling the fabrication processes from the precision requirements

of the bearing – In the 3DWN manufacturing process the errors associated with

the non-precision parts have less of an effect on bearing performance then errors

in the precision parts. This is powerful, in that it supports the feasibility of 3DWN

technology by showing a hydrostatic bearing’s precision requirements can be

decoupled from the fabrication processes used to make it.

9. Metrics to judge the required precision of off-the-shelf parts used in 3DWN

technology – Metrics to judge the level of precision required in the off-the-shelf

parts used to make a 3DWN bearing are defined from conventional hydrostatic

bearing theory.

10. A new process of manufacturing parts with internal features – 3DWN is a new

process of making parts with internal features. 3DWN is not limited to fluid

channels; it could be used for many kinds of applications that require cylindrical

internal networks.

11. A model to describe a wrapped structure – A model for describing a wrapped

structure is derived. This model enables coordinates in a 3D cylindrical structure

to be transformed to a 2D Template. When the Template is rolled, the 2D features

align in the wrapped structure to recreate the original 3D features.

12. A means of converting 3D bearing geometry to 2D features – Existing bearing

features are modified such that it can be cut into a 2D Template. Conventional

hydrostatic bearing analysis is used to evaluate the bearing performance.

7.2 Engineering impact The results of this thesis may impact engineering applications in the following ways:

1. An order of magnitude cost reduction from current bearings – The cost

comparison in this thesis demonstrates a 10X reduction in cost from a babbitted

bearing to a 3DWN bearing of the same sized and topology. Reducing the cost of

hydrostatic bearings by an order of magnitude would facilitate their use in more

engineering applications.

2. A flexible process in which to design and fabricate hydrostatic bearings – 3DWN

technology could provide great flexibility in manufacturing and design of custom

hydrostatic bearings. When the 3DWN manufacturing process is perfected, a

flexible bearing fabrication machine could be designed which makes any size

bearing by simply switching the mandrel size. As a result hydrostatic bearings

would not be “one-off” designs, which would further lower the cost of production.

3. A hybridization between 3DWN and current bearing technology – 3DWN

technology is an ideal method of producing self-compensated bearings because

the internal channels are cut at the same time as the bearing topology. As was

seen in the cost comparison in Chapter 6, the major cost in producing the

babbitted bearing was the manufacturing internal channels. 3DWN-conventional

hybrid bearings could possibly be produced by wrapping a template around a

cylinder that already has the required bore precision, for example a bronze

bushing.

4. A low-cost method of producing precision bores – Precision bore wrapping could

have other applications beyond hydrostatic bearings. One possible application

could be in hydrodynamic bearings. The overlap region of a 3DWN structure

could stagnate the gas flow in the bearing, causing a high pressure zone. The

overlap region could also be used as a feed for lubrication, similar to the notches

in hydrodynamic bearings found in IC engines.

5. Expansion of 3DWN technology to other products – 3DWN technology could be

expanded to other applications that require internal networks. Some of these

applications, currently being explored by the author, are fuel cells and circuit

boards. The cylindrical shape of 3DWN structures makes them a possibility for

use in applications such as batteries, missiles, or medical implants in the

circulatory system.

7.3 Future work The following areas of research would increase the feasibility of hydrostatic bearings

being made using 3DWN technology.

1. Possible adhesives for bonding wrapped layers needs to be further researched. As

was described in Chapter 4, many methods of adhering the HBP were attempted

without success. Adhesive films still may be a possible solution with

parameterization of their flow characteristics.

2. Most of the design effort in the HBP was focused on the topology and not the

internal features. Internal feature layout could be optimized to insure better

adhesion between layers, which would reduce the likelihood of leaks.

3. Alternative cutting processes should be explored beyond waterjet cutting. Laser

cutting is attractive because 1) it is low cost 2) it is faster than waterjet cutting, 3)

it doesn’t require protective layers of plastic sandwiching the template, and 4) it

leaves a bur on only one side. Inert environment laser cutting may be an option

for manufacturing bronze Templates.

4. To save manual labor time, chemical deburring and etching processes should be

explored. To enable chemical deburring, the Template could be cut with a plastic

film covering the bearing surface. When dipped into the chemical bath, the plastic

film would protect the bearing topology from becoming etched.

5. Better methods of fixturing the Template to the mandrel should be explored. The

hose clamps scratched the surface of the mandrel during fabrication of the HBP,

which is undesirable for mandrel re-use in multiple bearings. Also, the sacrificial

tabs have to be carefully removed, as to not damage the bore. Possible methods of

fixturing could include vacuuming the Template in place, or loosely wrapping

multiple layers and using a capstan effect to tighten them against the mandrel.

REFERENCES [1] Slocum, Alexander H. Precision Machine Design. Dearborn: Society of

Manufacturing Engineers, 1992. [2] Bassani, R., Piccigallo, B., Hydrostatic Lubrication. Amsterdam: Elsevier, 1992. [3] Geary, P. G. Fluid Film Bearings. Kent: British Scientific Instrument Research

Association, 1962. [4] Kotilainen, M.S.A. Design and Manufacturing of Modular Self-Compensating

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